diff --git "a/parse/dev/9XFSbDPmdW/9XFSbDPmdW_middle.json" "b/parse/dev/9XFSbDPmdW/9XFSbDPmdW_middle.json" new file mode 100644--- /dev/null +++ "b/parse/dev/9XFSbDPmdW/9XFSbDPmdW_middle.json" @@ -0,0 +1,83886 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 80, + 420, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 420, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 420, + 98 + ], + "score": 1.0, + "content": "PROGRESS MEASURES FOR GROKKING VIA", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 102, + 357, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 357, + 117 + ], + "score": 1.0, + "content": "MECHANISTIC INTERPRETABILITY", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 115, + 133, + 496, + 146 + ], + "lines": [ + { + "bbox": [ + 113, + 132, + 498, + 149 + ], + "spans": [ + { + "bbox": [ + 113, + 132, + 498, + 149 + ], + "score": 1.0, + "content": "Neel Nanda∗, † Lawrence Chan‡ Tom Lieberum† Jess Smith† Jacob Steinhardt‡", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 278, + 175, + 333, + 187 + ], + "lines": [ + { + "bbox": [ + 276, + 174, + 335, + 189 + ], + "spans": [ + { + "bbox": [ + 276, + 174, + 335, + 189 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 143, + 199, + 468, + 385 + ], + "lines": [ + { + "bbox": [ + 141, + 198, + 469, + 213 + ], + "spans": [ + { + "bbox": [ + 141, + 198, + 469, + 213 + ], + "score": 1.0, + "content": "Neural networks often exhibit emergent behavior, where qualitatively new capa-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 141, + 209, + 469, + 224 + ], + "spans": [ + { + "bbox": [ + 141, + 209, + 469, + 224 + ], + "score": 1.0, + "content": "bilities arise from scaling up the amount of parameters, training data, or training", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 222, + 469, + 234 + ], + "spans": [ + { + "bbox": [ + 141, + 222, + 469, + 234 + ], + "score": 1.0, + "content": "steps. 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This has led to a number of breakthroughs, via capabilities such as in-context learning (Rad-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "ford et al., 2019; Brown et al., 2020) and chain-of-thought prompting (Wei et al., 2022b). However,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "it also poses risks: Pan et al. 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We could better", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 524, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 536 + ], + "score": 1.0, + "content": "understand and predict these phase transitions by finding hidden progress measures (Barak et al.,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 535, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 505, + 546 + ], + "score": 1.0, + "content": "2022): metrics that precede and are causally linked to the phase transition, and which vary more", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "smoothly. For example, Wei et al. 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Small transformers trained", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 661, + 454, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 661, + 454, + 673 + ], + "score": 1.0, + "content": "with weight decay on this task consistently exhibit grokking (Figure 2, Appendix C.2).", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5 + } + ], + "page_idx": 0, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 679, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 118, + 677, + 440, + 690 + ], + "spans": [ + { + "bbox": [ + 118, + 677, + 440, + 690 + ], + "score": 1.0, + "content": "∗Corresponding author, please direct correspondence to: neelnanda27@gmail.com", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 686, + 210, + 703 + ], + "spans": [ + { + "bbox": [ + 118, + 686, + 210, + 703 + ], + "score": 1.0, + "content": "†Independent researcher.", + "type": "text" + } + ] + }, + { + "bbox": [ + 117, + 697, + 250, + 714 + ], + "spans": [ + { + "bbox": [ + 117, + 697, + 250, + 714 + ], + "score": 1.0, + "content": "‡University of California, Berkeley.", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 708, + 505, + 725 + ], + "spans": [ + { + "bbox": [ + 118, + 708, + 505, + 725 + ], + "score": 1.0, + "content": "1Interactive versions of figures, as well as the code to reproduce our results, are available at bit.ly/", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 295, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 295, + 733 + ], + "score": 1.0, + "content": "progress-measures-grokking-website.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "1", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 80, + 420, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 420, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 420, + 98 + ], + "score": 1.0, + "content": "PROGRESS MEASURES FOR GROKKING VIA", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 102, + 357, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 357, + 117 + ], + "score": 1.0, + "content": "MECHANISTIC INTERPRETABILITY", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 115, + 133, + 496, + 146 + ], + "lines": [ + { + "bbox": [ + 113, + 132, + 498, + 149 + ], + "spans": [ + { + "bbox": [ + 113, + 132, + 498, + 149 + ], + "score": 1.0, + "content": "Neel Nanda∗, † Lawrence Chan‡ Tom Lieberum† Jess Smith† Jacob Steinhardt‡", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 113, + 132, + 498, + 149 + ] + }, + { + "type": "title", + "bbox": [ + 278, + 175, + 333, + 187 + ], + "lines": [ + { + "bbox": [ + 276, + 174, + 335, + 189 + ], + "spans": [ + { + "bbox": [ + 276, + 174, + 335, + 189 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 143, + 199, + 468, + 385 + ], + "lines": [ + { + "bbox": [ + 141, + 198, + 469, + 213 + ], + "spans": [ + { + "bbox": [ + 141, + 198, + 469, + 213 + ], + "score": 1.0, + "content": "Neural networks often exhibit emergent behavior, where qualitatively new capa-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 141, + 209, + 469, + 224 + ], + "spans": [ + { + "bbox": [ + 141, + 209, + 469, + 224 + ], + "score": 1.0, + "content": "bilities arise from scaling up the amount of parameters, training data, or training", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 222, + 469, + 234 + ], + "spans": [ + { + "bbox": [ + 141, + 222, + 469, + 234 + ], + "score": 1.0, + "content": "steps. 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Based on this understanding, we define progress", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 330, + 470, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 330, + 470, + 344 + ], + "score": 1.0, + "content": "measures that allow us to study the dynamics of training and split training into", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 343, + 469, + 354 + ], + "spans": [ + { + "bbox": [ + 142, + 343, + 469, + 354 + ], + "score": 1.0, + "content": "three continuous phases: memorization, circuit formation, and cleanup. Our re-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 353, + 469, + 365 + ], + "spans": [ + { + "bbox": [ + 142, + 353, + 469, + 365 + ], + "score": 1.0, + "content": "sults show that grokking, rather than being a sudden shift, arises from the gradual", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 363, + 469, + 377 + ], + "spans": [ + { + "bbox": [ + 141, + 363, + 469, + 377 + ], + "score": 1.0, + "content": "amplification of structured mechanisms encoded in the weights, followed by the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 374, + 312, + 387 + ], + "spans": [ + { + "bbox": [ + 141, + 374, + 312, + 387 + ], + "score": 1.0, + "content": "later removal of memorizing components.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 12, + "bbox_fs": [ + 141, + 198, + 470, + 387 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 405, + 206, + 418 + ], + "lines": [ + { + "bbox": [ + 105, + 404, + 208, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 208, + 421 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 430, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 430, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 441 + ], + "score": 1.0, + "content": "Neural networks often exhibit emergent behavior, in which qualitatively new capabilities arise from", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "scaling up the model size, training data, or number of training steps (Steinhardt, 2022; Wei et al.,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "2022a). This has led to a number of breakthroughs, via capabilities such as in-context learning (Rad-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "ford et al., 2019; Brown et al., 2020) and chain-of-thought prompting (Wei et al., 2022b). However,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "it also poses risks: Pan et al. (2022) show that scaling up the parameter count of models by as little", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 484, + 289, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 117, + 498 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 117, + 485, + 137, + 495 + ], + "score": 0.85, + "content": "30 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 484, + 289, + 498 + ], + "score": 1.0, + "content": "can lead to emergent reward hacking.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 430, + 505, + 498 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 501, + 505, + 579 + ], + "lines": [ + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "Emergence is most surprising when it is abrupt, as in the case of reward hacking, chain-of-thought", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 512, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 524 + ], + "score": 1.0, + "content": "reasoning, or other phase transitions (Ganguli et al., 2022; Wei et al., 2022a). We could better", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 524, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 536 + ], + "score": 1.0, + "content": "understand and predict these phase transitions by finding hidden progress measures (Barak et al.,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 535, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 505, + 546 + ], + "score": 1.0, + "content": "2022): metrics that precede and are causally linked to the phase transition, and which vary more", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "smoothly. For example, Wei et al. (2022a) show that while large language models show abrupt", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 557, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 505, + 568 + ], + "score": 1.0, + "content": "jumps in their performance on many benchmarks, their cross-entropy loss decreases smoothly with", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 567, + 450, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 450, + 581 + ], + "score": 1.0, + "content": "model scale. However, cross-entropy does not explain why the phase changes happen.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 501, + 505, + 581 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 584, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 597 + ], + "score": 1.0, + "content": "In this work, we introduce a different approach to uncovering hidden progress measures: via mech-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "anistic explanations.1 A mechanistic explanation aims to reverse engineer the mechanisms of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "score": 1.0, + "content": "network, generally by identifying the circuits (Cammarata et al., 2020; Elhage et al., 2021) within", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 617, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 630 + ], + "score": 1.0, + "content": "a model that implement a behavior. Using such explanations, we study grokking, where models", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "abruptly transition to a generalizing solution after a large number of training steps, despite initially", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "overfitting (Power et al., 2022). Specifically, we study modular addition, where a model takes inputs", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 650, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 107, + 650, + 194, + 662 + ], + "score": 0.93, + "content": "a , b \\in \\{ 0 , \\ldots , P - 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 650, + 257, + 663 + ], + "score": 1.0, + "content": "for some prime", + "type": "text" + }, + { + "bbox": [ + 258, + 651, + 267, + 660 + ], + "score": 0.82, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 650, + 357, + 663 + ], + "score": 1.0, + "content": "and predicts their sum", + "type": "text" + }, + { + "bbox": [ + 357, + 652, + 363, + 660 + ], + "score": 0.54, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 650, + 383, + 663 + ], + "score": 1.0, + "content": "mod", + "type": "text" + }, + { + "bbox": [ + 384, + 651, + 393, + 660 + ], + "score": 0.77, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 650, + 505, + 663 + ], + "score": 1.0, + "content": ". Small transformers trained", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 661, + 454, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 661, + 454, + 673 + ], + "score": 1.0, + "content": "with weight decay on this task consistently exhibit grokking (Figure 2, Appendix C.2).", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 583, + 505, + 673 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 124, + 56, + 483, + 199 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 124, + 56, + 483, + 199 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 124, + 56, + 483, + 199 + ], + "spans": [ + { + "bbox": [ + 124, + 56, + 483, + 199 + ], + "score": 0.96, + "type": "image", + "image_path": "371d8e06d208cbc56f63ab4815491296c5e8e01d876de7d0d128f91d1ffc58d4.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 124, + 56, + 483, + 103.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 124, + 103.66666666666666, + 483, + 151.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 124, + 151.33333333333331, + 483, + 198.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 205, + 505, + 261 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 505, + 217 + ], + "score": 1.0, + "content": "Figure 1: The algorithm implemented by the one-layer transformer for modular addition. Given two", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 215, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 143, + 229 + ], + "score": 1.0, + "content": "numbers", + "type": "text" + }, + { + "bbox": [ + 144, + 218, + 150, + 226 + ], + "score": 0.72, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 215, + 169, + 229 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 169, + 217, + 174, + 226 + ], + "score": 0.72, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 215, + 505, + 229 + ], + "score": 1.0, + "content": ", the model projects each point onto a corresponding rotation using its embedding", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 226, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 240 + ], + "score": 1.0, + "content": "matrix. Using its attention and MLP layers, it then composes the rotations to get a representation of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 237, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 130, + 249 + ], + "score": 0.89, + "content": "a + b", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 237, + 157, + 251 + ], + "score": 1.0, + "content": "mod", + "type": "text" + }, + { + "bbox": [ + 158, + 238, + 167, + 248 + ], + "score": 0.74, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 237, + 333, + 251 + ], + "score": 1.0, + "content": ". Finally, it “reads off” the logits for each", + "type": "text" + }, + { + "bbox": [ + 333, + 238, + 417, + 250 + ], + "score": 0.92, + "content": "c \\in \\{ 0 , 1 , . . . , P - 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 237, + 480, + 251 + ], + "score": 1.0, + "content": ", by rotating by", + "type": "text" + }, + { + "bbox": [ + 480, + 239, + 494, + 248 + ], + "score": 0.74, + "content": "- c", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 237, + 506, + 251 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 247, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 121, + 263 + ], + "score": 1.0, + "content": "get", + "type": "text" + }, + { + "bbox": [ + 122, + 249, + 196, + 261 + ], + "score": 0.89, + "content": "\\cos ( w ( a + b - c ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 247, + 307, + 263 + ], + "score": 1.0, + "content": ", which is maximized when", + "type": "text" + }, + { + "bbox": [ + 308, + 249, + 349, + 259 + ], + "score": 0.89, + "content": "a + b \\equiv c", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 247, + 375, + 263 + ], + "score": 1.0, + "content": "mod", + "type": "text" + }, + { + "bbox": [ + 376, + 250, + 385, + 259 + ], + "score": 0.81, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 247, + 412, + 263 + ], + "score": 1.0, + "content": "(since", + "type": "text" + }, + { + "bbox": [ + 412, + 251, + 421, + 259 + ], + "score": 0.76, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 247, + 485, + 263 + ], + "score": 1.0, + "content": "is a multiple of", + "type": "text" + }, + { + "bbox": [ + 486, + 248, + 497, + 262 + ], + "score": 0.89, + "content": "\\frac { 2 \\pi } { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 498, + 247, + 506, + 263 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 288, + 504, + 343 + ], + "lines": [ + { + "bbox": [ + 106, + 288, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 505, + 302 + ], + "score": 1.0, + "content": "We reverse engineer the weights of these transformers and find that they perform this task by map-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "score": 1.0, + "content": "ping the inputs onto a circle and performing addition on the circle. Specifically, we show that the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 308, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 104, + 308, + 250, + 325 + ], + "score": 1.0, + "content": "embedding matrix maps the inputs", + "type": "text" + }, + { + "bbox": [ + 250, + 311, + 266, + 322 + ], + "score": 0.88, + "content": "a , b", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 308, + 488, + 325 + ], + "score": 1.0, + "content": "to sines and cosines at a sparse set of key frequencies", + "type": "text" + }, + { + "bbox": [ + 488, + 312, + 501, + 321 + ], + "score": 0.84, + "content": "w _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 308, + 506, + 325 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 320, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 334 + ], + "score": 1.0, + "content": "The attention and MLP layers then combine these using trigonometric identities to compute the sine", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 332, + 452, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 162, + 345 + ], + "score": 1.0, + "content": "and cosine of", + "type": "text" + }, + { + "bbox": [ + 162, + 332, + 204, + 344 + ], + "score": 0.93, + "content": "w _ { k } ( a + b )", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 332, + 452, + 345 + ], + "score": 1.0, + "content": ", and the output matrices shift and combine these frequencies.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 349, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 106, + 348, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 505, + 362 + ], + "score": 1.0, + "content": "We confirm this understanding with four lines of evidence (Section 4): (1) the network weights", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 424, + 372 + ], + "score": 1.0, + "content": "and activations exhibit a consistent periodic structure; (2) the neuron-logit map", + "type": "text" + }, + { + "bbox": [ + 425, + 360, + 441, + 371 + ], + "score": 0.88, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "is well approx-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 370, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 384 + ], + "score": 1.0, + "content": "imated by a sum of sinusoidal functions of the key frequencies, and projecting the MLP activations", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 383, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 505, + 394 + ], + "score": 1.0, + "content": "onto these sinusoidal functions lets us “read off” trigonometric identities from the neurons; (3) the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "attention heads and MLP neuron are well approximated by degree-2 polynomials of trigonometric", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "functions of a single frequency; and (4) ablating key frequencies used by the model reduces perfor-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 415, + 480, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 274, + 428 + ], + "score": 1.0, + "content": "mance to chance, while ablating the other", + "type": "text" + }, + { + "bbox": [ + 275, + 415, + 294, + 425 + ], + "score": 0.87, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 415, + 480, + 428 + ], + "score": 1.0, + "content": "of frequencies slightly improves performance.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 432, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 105, + 431, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 505, + 445 + ], + "score": 1.0, + "content": "Using our understanding of the learned algorithm, we construct two progress measures for the mod-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "score": 1.0, + "content": "ular addition task—restricted loss, where we ablate every non-key frequency, and excluded loss,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 454, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 466 + ], + "score": 1.0, + "content": "where we instead ablate all key frequencies. Both metrics improve continuously prior to when", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 464, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 478 + ], + "score": 1.0, + "content": "grokking occurs. We use these metrics to understand the training dynamics underlying grokking", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 476, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 506, + 488 + ], + "score": 1.0, + "content": "and find that training can be split into three phases: memorization of the training data; circuit for-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "mation, where the network learns a mechanism that generalizes; and cleanup, where weight decay", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "score": 1.0, + "content": "removes the memorization components. Surprisingly, the sudden transition to perfect test accuracy", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 509, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 521 + ], + "score": 1.0, + "content": "in grokking occurs during cleanup, after the generalizing mechanism is learned. These results show", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "that grokking, rather than being a sudden shift, arises from the gradual amplification of structured", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 530, + 489, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 489, + 543 + ], + "score": 1.0, + "content": "mechanisms encoded in the weights, followed by the later removal of memorizing components.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 108, + 564, + 210, + 577 + ], + "lines": [ + { + "bbox": [ + 104, + 563, + 213, + 580 + ], + "spans": [ + { + "bbox": [ + 104, + 563, + 213, + 580 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 504, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "Phase Changes. Recent papers have observed that neural networks quickly develop novel quali-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "tative behaviors as they are scaled up or trained longer (Ganguli et al., 2022; Wei et al., 2022a).", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 615, + 504, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 488, + 628 + ], + "score": 1.0, + "content": "McGrath et al. (2021) find that AlphaZero quickly learns many human chess concepts between", + "type": "text" + }, + { + "bbox": [ + 488, + 616, + 504, + 627 + ], + "score": 0.31, + "content": "1 0 \\mathrm { k }", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 626, + 490, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 123, + 640 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 627, + 140, + 637 + ], + "score": 0.37, + "content": "3 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 626, + 379, + 640 + ], + "score": 1.0, + "content": "training steps and reinvents human opening theory between", + "type": "text" + }, + { + "bbox": [ + 380, + 627, + 397, + 637 + ], + "score": 0.28, + "content": "2 5 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 626, + 490, + 640 + ], + "score": 1.0, + "content": "and 60k training steps.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 657 + ], + "score": 1.0, + "content": "Grokking. Grokking was first reported in Power et al. (2022), which trained two-layer transformers", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "on several algorithmic tasks and found that test accuracy often increased sharply long after achieving", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "perfect train accuracy. Millidge (2022) suggests that this may be due to SGD being a random walk", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "on the optimal manifold. Our results echo Barak et al. (2022) in showing that the network instead", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "makes continuous progress toward the generalizing algorithm. Liu et al. (2022) construct small", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "examples of grokking, which they use to compute phase diagrams with four separate “phases” of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "learning. Thilak et al. 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Given two", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 215, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 143, + 229 + ], + "score": 1.0, + "content": "numbers", + "type": "text" + }, + { + "bbox": [ + 144, + 218, + 150, + 226 + ], + "score": 0.72, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 215, + 169, + 229 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 169, + 217, + 174, + 226 + ], + "score": 0.72, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 215, + 505, + 229 + ], + "score": 1.0, + "content": ", the model projects each point onto a corresponding rotation using its embedding", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 226, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 240 + ], + "score": 1.0, + "content": "matrix. 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Finally, it “reads off” the logits for each", + "type": "text" + }, + { + "bbox": [ + 333, + 238, + 417, + 250 + ], + "score": 0.92, + "content": "c \\in \\{ 0 , 1 , . . . , P - 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 237, + 480, + 251 + ], + "score": 1.0, + "content": ", by rotating by", + "type": "text" + }, + { + "bbox": [ + 480, + 239, + 494, + 248 + ], + "score": 0.74, + "content": "- c", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 237, + 506, + 251 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 247, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 121, + 263 + ], + "score": 1.0, + "content": "get", + "type": "text" + }, + { + "bbox": [ + 122, + 249, + 196, + 261 + ], + "score": 0.89, + "content": "\\cos ( w ( a + b - c ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 247, + 307, + 263 + ], + "score": 1.0, + "content": ", which is maximized when", + "type": "text" + }, + { + "bbox": [ + 308, + 249, + 349, + 259 + ], + "score": 0.89, + "content": "a + b \\equiv c", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 247, + 375, + 263 + ], + "score": 1.0, + "content": "mod", + "type": "text" + }, + { + "bbox": [ + 376, + 250, + 385, + 259 + ], + "score": 0.81, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 247, + 412, + 263 + ], + "score": 1.0, + "content": "(since", + "type": "text" + }, + { + "bbox": [ + 412, + 251, + 421, + 259 + ], + "score": 0.76, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 247, + 485, + 263 + ], + "score": 1.0, + "content": "is a multiple of", + "type": "text" + }, + { + "bbox": [ + 486, + 248, + 497, + 262 + ], + "score": 0.89, + "content": "\\frac { 2 \\pi } { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 498, + 247, + 506, + 263 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 288, + 504, + 343 + ], + "lines": [ + { + "bbox": [ + 106, + 288, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 505, + 302 + ], + "score": 1.0, + "content": "We reverse engineer the weights of these transformers and find that they perform this task by map-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "score": 1.0, + "content": "ping the inputs onto a circle and performing addition on the circle. Specifically, we show that the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 308, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 104, + 308, + 250, + 325 + ], + "score": 1.0, + "content": "embedding matrix maps the inputs", + "type": "text" + }, + { + "bbox": [ + 250, + 311, + 266, + 322 + ], + "score": 0.88, + "content": "a , b", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 308, + 488, + 325 + ], + "score": 1.0, + "content": "to sines and cosines at a sparse set of key frequencies", + "type": "text" + }, + { + "bbox": [ + 488, + 312, + 501, + 321 + ], + "score": 0.84, + "content": "w _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 308, + 506, + 325 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 320, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 334 + ], + "score": 1.0, + "content": "The attention and MLP layers then combine these using trigonometric identities to compute the sine", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 332, + 452, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 162, + 345 + ], + "score": 1.0, + "content": "and cosine of", + "type": "text" + }, + { + "bbox": [ + 162, + 332, + 204, + 344 + ], + "score": 0.93, + "content": "w _ { k } ( a + b )", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 332, + 452, + 345 + ], + "score": 1.0, + "content": ", and the output matrices shift and combine these frequencies.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10, + "bbox_fs": [ + 104, + 288, + 506, + 345 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 349, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 106, + 348, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 505, + 362 + ], + "score": 1.0, + "content": "We confirm this understanding with four lines of evidence (Section 4): (1) the network weights", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 424, + 372 + ], + "score": 1.0, + "content": "and activations exhibit a consistent periodic structure; 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and (4) ablating key frequencies used by the model reduces perfor-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 415, + 480, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 274, + 428 + ], + "score": 1.0, + "content": "mance to chance, while ablating the other", + "type": "text" + }, + { + "bbox": [ + 275, + 415, + 294, + 425 + ], + "score": 0.87, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 415, + 480, + 428 + ], + "score": 1.0, + "content": "of frequencies slightly improves performance.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 348, + 505, + 428 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 432, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 105, + 431, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 505, + 445 + ], + "score": 1.0, + "content": "Using our understanding of the learned algorithm, we construct two progress measures for the mod-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "score": 1.0, + "content": "ular addition task—restricted loss, where we ablate every non-key frequency, and excluded loss,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 454, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 466 + ], + "score": 1.0, + "content": "where we instead ablate all key frequencies. Both metrics improve continuously prior to when", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 464, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 478 + ], + "score": 1.0, + "content": "grokking occurs. We use these metrics to understand the training dynamics underlying grokking", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 476, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 506, + 488 + ], + "score": 1.0, + "content": "and find that training can be split into three phases: memorization of the training data; circuit for-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "mation, where the network learns a mechanism that generalizes; and cleanup, where weight decay", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "score": 1.0, + "content": "removes the memorization components. 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These results show", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "that grokking, rather than being a sudden shift, arises from the gradual amplification of structured", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 530, + 489, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 489, + 543 + ], + "score": 1.0, + "content": "mechanisms encoded in the weights, followed by the later removal of memorizing components.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 431, + 506, + 543 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 564, + 210, + 577 + ], + "lines": [ + { + "bbox": [ + 104, + 563, + 213, + 580 + ], + "spans": [ + { + "bbox": [ + 104, + 563, + 213, + 580 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 504, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "Phase Changes. Recent papers have observed that neural networks quickly develop novel quali-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "tative behaviors as they are scaled up or trained longer (Ganguli et al., 2022; Wei et al., 2022a).", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 615, + 504, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 488, + 628 + ], + "score": 1.0, + "content": "McGrath et al. 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Grokking was first reported in Power et al. (2022), which trained two-layer transformers", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "on several algorithmic tasks and found that test accuracy often increased sharply long after achieving", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "perfect train accuracy. Millidge (2022) suggests that this may be due to SGD being a random walk", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "on the optimal manifold. Our results echo Barak et al. 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These models consistently", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 178, + 473, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 473, + 190 + ], + "score": 1.0, + "content": "exhibit grokking: they quickly overfit early on in training, but then later learn to generalize.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 216, + 505, + 249 + ], + "lines": [ + { + "bbox": [ + 106, + 215, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 505, + 228 + ], + "score": 1.0, + "content": "Circuits-style mechanistic interpretability. The style of post-hoc mechanistic interpretability in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "score": 1.0, + "content": "Section 4 is heavily inspired by the Circuits approach of Cammarata et al. 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They prove theoretically that training would", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 290 + ], + "score": 1.0, + "content": "amplify a certain mechanism and heuristically define a progress measure. 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In other experiments, we vary the depth", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 404, + 480, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 480, + 416 + ], + "score": 1.0, + "content": "and dimension of the model. 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As there are only", + "type": "text" + }, + { + "bbox": [ + 322, + 454, + 361, + 464 + ], + "score": 0.5, + "content": "1 1 3 \\cdot 1 1 3", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "possible pairs, we evaluate test loss", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 464, + 331, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 331, + 478 + ], + "score": 1.0, + "content": "and accuracy on all pairs of inputs not used for training.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "Networks trained on this task consistently exhibit grokking. As Figure 2 shows, our networks first", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 341, + 505 + ], + "score": 1.0, + "content": "overfit the training set: train accuracy quickly converges to", + "type": "text" + }, + { + "bbox": [ + 342, + 493, + 365, + 503 + ], + "score": 0.88, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "and the train loss quickly declines,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 440, + 516 + ], + "score": 1.0, + "content": "while the test accuracy remains low and the test loss remains high. After around", + "type": "text" + }, + { + "bbox": [ + 440, + 504, + 471, + 515 + ], + "score": 0.25, + "content": "1 0 , 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "epochs,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 349, + 527 + ], + "score": 1.0, + "content": "the network generalizes and test accuracy increases to near", + "type": "text" + }, + { + "bbox": [ + 349, + 515, + 374, + 525 + ], + "score": 0.88, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 514, + 506, + 527 + ], + "score": 1.0, + "content": ". 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After around", + "type": "text" + }, + { + "bbox": [ + 440, + 504, + 471, + 515 + ], + "score": 0.25, + "content": "1 0 , 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "epochs,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 349, + 527 + ], + "score": 1.0, + "content": "the network generalizes and test accuracy increases to near", + "type": "text" + }, + { + "bbox": [ + 349, + 515, + 374, + 525 + ], + "score": 0.88, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 514, + 506, + 527 + ], + "score": 1.0, + "content": ". 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Here we apply our analysis to the mainline model", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "from Section 3; the results are broadly consistent for other models, including across different num-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 310, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 505, + 324 + ], + "score": 1.0, + "content": "ber of layers, different fractions of the training data, and different prime moduli (see Appendix C.2,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 321, + 187, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 187, + 335 + ], + "score": 1.0, + "content": "especially Table 5).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 339, + 505, + 383 + ], + "lines": [ + { + "bbox": [ + 105, + 338, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 352 + ], + "score": 1.0, + "content": "Our first line of evidence involves examining the network weights and activations and observing", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 350, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 505, + 362 + ], + "score": 1.0, + "content": "consistent periodic structure that is unlikely to occur by chance (Section 4.1). Moreover, when we", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "take Fourier transforms, many components are either sparse or nearly sparse in the Fourier domain,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 372, + 277, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 277, + 385 + ], + "score": 1.0, + "content": "supported on a handful of key frequencies.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 389, + 505, + 444 + ], + "lines": [ + { + "bbox": [ + 107, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 107, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "We next look into the actual mechanisms implemented in the model weights (Section 4.2). We show", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 399, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 225, + 411 + ], + "score": 1.0, + "content": "that the unembedding matrix", + "type": "text" + }, + { + "bbox": [ + 226, + 400, + 242, + 411 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 399, + 505, + 411 + ], + "score": 1.0, + "content": "is (approximately) rank 10, where each direction corresponds to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "the cosine or sine of one of 5 key frequencies. Projecting the MLP activations onto the components", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 420, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 118, + 435 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 119, + 422, + 135, + 433 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 420, + 350, + 435 + ], + "score": 1.0, + "content": "approximately produces multiples of the functions", + "type": "text" + }, + { + "bbox": [ + 351, + 421, + 415, + 434 + ], + "score": 0.91, + "content": "\\cos { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 420, + 437, + 435 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 437, + 421, + 500, + 434 + ], + "score": 0.92, + "content": "\\sin { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 420, + 505, + 435 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 432, + 326, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 326, + 446 + ], + "score": 1.0, + "content": "showing that the MLP layer does compute these sums.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 449, + 505, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 449, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 505, + 461 + ], + "score": 1.0, + "content": "To better understand the mechanism, we zoom in to individual neurons (Section 4.3). We find that", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "the attention heads and most neurons are well-approximated by degree-2 polynomials of sines and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 471, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 394, + 483 + ], + "score": 1.0, + "content": "cosines at a single frequency. Moreover, the corresponding direction in", + "type": "text" + }, + { + "bbox": [ + 394, + 471, + 411, + 482 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 471, + 505, + 483 + ], + "score": 1.0, + "content": "also contains only that", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 482, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 505, + 496 + ], + "score": 1.0, + "content": "frequency. This suggests that the model’s computations are (1) localized across frequencies and (2)", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 494, + 258, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 258, + 506 + ], + "score": 1.0, + "content": "mostly aligned with the neuron basis.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 505, + 543 + ], + "lines": [ + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "Finally, we use ablations to confirm that our interpretation is faithful (Section 4.4). We replace", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 521, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 534 + ], + "score": 1.0, + "content": "various components of the model by the components of the Fourier multiplication algorithm and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 532, + 492, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 492, + 545 + ], + "score": 1.0, + "content": "find that doing so consistently does not harm and sometimes even improves model performance.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 106, + 557, + 349, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 556, + 350, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 350, + 569 + ], + "score": 1.0, + "content": "4.1 SUGGESTIVE EVIDENCE: SURPRISING PERIODICITY", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 108, + 577, + 503, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "The first line of evidence that the network is using the algorithm described in Section 3.1 is the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "surprising periodicity in the activations of the transformer. That is, the output of every part of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 599, + 319, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 319, + 612 + ], + "score": 1.0, + "content": "network is periodic as a function of the input tokens.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 615, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "Periodicity in the embeddings. We start by examining the embeddings. We apply a Fourier trans-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 338, + 640 + ], + "score": 1.0, + "content": "form along the input dimension of the embedding matrix", + "type": "text" + }, + { + "bbox": [ + 338, + 627, + 356, + 638 + ], + "score": 0.91, + "content": "W _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 626, + 429, + 640 + ], + "score": 1.0, + "content": "then compute the", + "type": "text" + }, + { + "bbox": [ + 429, + 627, + 439, + 638 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "-norm along the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "other dimension; results are shown in Figure 3. We plot only the components for the first 56 frequen-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 319, + 662 + ], + "score": 1.0, + "content": "cies, as the norm of the components for frequencies", + "type": "text" + }, + { + "bbox": [ + 320, + 649, + 327, + 659 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 649, + 345, + 662 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 346, + 649, + 374, + 659 + ], + "score": 0.91, + "content": "P - k", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "are symmetric. The embedding", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 135, + 673 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 135, + 660, + 152, + 671 + ], + "score": 0.9, + "content": "W _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "is sparse in the Fourier basis–it only has significant nonnegligible norm at 6 frequencies.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 670, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 685 + ], + "score": 1.0, + "content": "Of these frequencies, only 5 appear to be used significantly in later parts of the model (corresponding", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 681, + 410, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 116, + 694 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 681, + 215, + 694 + ], + "score": 0.89, + "content": "k \\in \\{ 1 4 , \\bar { 3 } 5 , 4 1 , 4 2 , 5 \\dot { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 681, + 410, + 694 + ], + "score": 1.0, + "content": "). We dub these the key frequencies of the model.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "Periodicity in attention heads and MLP neuron activations. This periodic structure recurs", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "throughout the network. As an example, we plot the attention weight at position 0 for every combi-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "score": 1.0, + "content": "nation of two inputs for head 0 in Figure 4. The attention exhibits a periodic structure with frequency", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "list", + "bbox": [ + 133, + 79, + 506, + 222 + ], + "lines": [ + { + "bbox": [ + 133, + 82, + 468, + 95 + ], + "spans": [ + { + "bbox": [ + 133, + 82, + 183, + 95 + ], + "score": 1.0, + "content": "• Compute", + "type": "text" + }, + { + "bbox": [ + 183, + 82, + 245, + 95 + ], + "score": 0.87, + "content": "\\cos { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 82, + 265, + 95 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 265, + 82, + 328, + 95 + ], + "score": 0.91, + "content": "\\sin { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 82, + 468, + 95 + ], + "score": 1.0, + "content": "using the trigonometric identities:", + "type": "text" + } + ], + "index": 0, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 196, + 96, + 450, + 124 + ], + "spans": [ + { + "bbox": [ + 196, + 96, + 450, + 124 + ], + "score": 0.89, + "content": "\\begin{array} { r l } & { \\cos { ( w _ { k } ( a + b ) ) } = \\cos { ( w _ { k } a ) } \\cos { ( w _ { k } a ) } - 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This", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 198, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 141, + 198, + 354, + 212 + ], + "score": 1.0, + "content": "causes the cosine waves to constructively interfere at", + "type": "text" + }, + { + "bbox": [ + 354, + 200, + 401, + 209 + ], + "score": 0.84, + "content": "c ^ { * } = a + b", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 198, + 426, + 212 + ], + "score": 1.0, + "content": "mod", + "type": "text" + }, + { + "bbox": [ + 426, + 200, + 433, + 210 + ], + "score": 0.78, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 198, + 464, + 212 + ], + "score": 1.0, + "content": "(giving", + "type": "text" + }, + { + "bbox": [ + 465, + 199, + 475, + 209 + ], + "score": 0.85, + "content": "c ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 198, + 505, + 212 + ], + "score": 1.0, + "content": "a large", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 210, + 491, + 223 + ], + "spans": [ + { + "bbox": [ + 141, + 210, + 474, + 223 + ], + "score": 1.0, + "content": "logit), and destructively interfere everywhere else (thus giving small logits to other", + "type": "text" + }, + { + "bbox": [ + 474, + 212, + 484, + 220 + ], + "score": 0.46, + "content": "c \\mathbf { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 210, + 491, + 223 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 9, + "is_list_end_line": true + } + ], + "index": 4.5, + "bbox_fs": [ + 132, + 82, + 506, + 223 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 226, + 503, + 239 + ], + "lines": [ + { + "bbox": [ + 107, + 226, + 504, + 239 + ], + "spans": [ + { + "bbox": [ + 107, + 226, + 504, + 239 + ], + "score": 1.0, + "content": "We refer to this algorithm as Fourier multiplication, and will justify our claim in detail in Section 4.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 107, + 226, + 504, + 239 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 254, + 402, + 266 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 404, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 404, + 268 + ], + "score": 1.0, + "content": "4 REVERSE ENGINEERING A ONE-LAYER TRANSFORMER", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 277, + 505, + 333 + ], + "lines": [ + { + "bbox": [ + 106, + 279, + 504, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 504, + 289 + ], + "score": 1.0, + "content": "In this section, we describe four lines of evidence that our transformers are using the Fourier mul-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "tiplication algorithm described in Section 3.1. Here we apply our analysis to the mainline model", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "from Section 3; the results are broadly consistent for other models, including across different num-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 310, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 505, + 324 + ], + "score": 1.0, + "content": "ber of layers, different fractions of the training data, and different prime moduli (see Appendix C.2,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 321, + 187, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 187, + 335 + ], + "score": 1.0, + "content": "especially Table 5).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 279, + 505, + 335 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 339, + 505, + 383 + ], + "lines": [ + { + "bbox": [ + 105, + 338, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 352 + ], + "score": 1.0, + "content": "Our first line of evidence involves examining the network weights and activations and observing", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 350, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 505, + 362 + ], + "score": 1.0, + "content": "consistent periodic structure that is unlikely to occur by chance (Section 4.1). Moreover, when we", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "take Fourier transforms, many components are either sparse or nearly sparse in the Fourier domain,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 372, + 277, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 277, + 385 + ], + "score": 1.0, + "content": "supported on a handful of key frequencies.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 338, + 505, + 385 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 389, + 505, + 444 + ], + "lines": [ + { + "bbox": [ + 107, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 107, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "We next look into the actual mechanisms implemented in the model weights (Section 4.2). We show", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 399, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 225, + 411 + ], + "score": 1.0, + "content": "that the unembedding matrix", + "type": "text" + }, + { + "bbox": [ + 226, + 400, + 242, + 411 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 399, + 505, + 411 + ], + "score": 1.0, + "content": "is (approximately) rank 10, where each direction corresponds to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "the cosine or sine of one of 5 key frequencies. Projecting the MLP activations onto the components", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 420, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 118, + 435 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 119, + 422, + 135, + 433 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 420, + 350, + 435 + ], + "score": 1.0, + "content": "approximately produces multiples of the functions", + "type": "text" + }, + { + "bbox": [ + 351, + 421, + 415, + 434 + ], + "score": 0.91, + "content": "\\cos { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 420, + 437, + 435 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 437, + 421, + 500, + 434 + ], + "score": 0.92, + "content": "\\sin { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 420, + 505, + 435 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 432, + 326, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 326, + 446 + ], + "score": 1.0, + "content": "showing that the MLP layer does compute these sums.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 389, + 505, + 446 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 449, + 505, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 449, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 505, + 461 + ], + "score": 1.0, + "content": "To better understand the mechanism, we zoom in to individual neurons (Section 4.3). We find that", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "the attention heads and most neurons are well-approximated by degree-2 polynomials of sines and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 471, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 394, + 483 + ], + "score": 1.0, + "content": "cosines at a single frequency. Moreover, the corresponding direction in", + "type": "text" + }, + { + "bbox": [ + 394, + 471, + 411, + 482 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 471, + 505, + 483 + ], + "score": 1.0, + "content": "also contains only that", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 482, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 505, + 496 + ], + "score": 1.0, + "content": "frequency. This suggests that the model’s computations are (1) localized across frequencies and (2)", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 494, + 258, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 258, + 506 + ], + "score": 1.0, + "content": "mostly aligned with the neuron basis.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 449, + 505, + 506 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 505, + 543 + ], + "lines": [ + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "Finally, we use ablations to confirm that our interpretation is faithful (Section 4.4). We replace", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 521, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 534 + ], + "score": 1.0, + "content": "various components of the model by the components of the Fourier multiplication algorithm and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 532, + 492, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 492, + 545 + ], + "score": 1.0, + "content": "find that doing so consistently does not harm and sometimes even improves model performance.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 510, + 505, + 545 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 557, + 349, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 556, + 350, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 350, + 569 + ], + "score": 1.0, + "content": "4.1 SUGGESTIVE EVIDENCE: SURPRISING PERIODICITY", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 108, + 577, + 503, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "The first line of evidence that the network is using the algorithm described in Section 3.1 is the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "surprising periodicity in the activations of the transformer. That is, the output of every part of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 599, + 319, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 319, + 612 + ], + "score": 1.0, + "content": "network is periodic as a function of the input tokens.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 106, + 577, + 505, + 612 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 615, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "Periodicity in the embeddings. We start by examining the embeddings. We apply a Fourier trans-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 338, + 640 + ], + "score": 1.0, + "content": "form along the input dimension of the embedding matrix", + "type": "text" + }, + { + "bbox": [ + 338, + 627, + 356, + 638 + ], + "score": 0.91, + "content": "W _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 626, + 429, + 640 + ], + "score": 1.0, + "content": "then compute the", + "type": "text" + }, + { + "bbox": [ + 429, + 627, + 439, + 638 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "-norm along the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "other dimension; results are shown in Figure 3. We plot only the components for the first 56 frequen-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 319, + 662 + ], + "score": 1.0, + "content": "cies, as the norm of the components for frequencies", + "type": "text" + }, + { + "bbox": [ + 320, + 649, + 327, + 659 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 649, + 345, + 662 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 346, + 649, + 374, + 659 + ], + "score": 0.91, + "content": "P - k", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "are symmetric. The embedding", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 135, + 673 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 135, + 660, + 152, + 671 + ], + "score": 0.9, + "content": "W _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "is sparse in the Fourier basis–it only has significant nonnegligible norm at 6 frequencies.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 670, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 685 + ], + "score": 1.0, + "content": "Of these frequencies, only 5 appear to be used significantly in later parts of the model (corresponding", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 681, + 410, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 116, + 694 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 681, + 215, + 694 + ], + "score": 0.89, + "content": "k \\in \\{ 1 4 , \\bar { 3 } 5 , 4 1 , 4 2 , 5 \\dot { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 681, + 410, + 694 + ], + "score": 1.0, + "content": "). We dub these the key frequencies of the model.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 615, + 505, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "Periodicity in attention heads and MLP neuron activations. This periodic structure recurs", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "throughout the network. As an example, we plot the attention weight at position 0 for every combi-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "score": 1.0, + "content": "nation of two inputs for head 0 in Figure 4. The attention exhibits a periodic structure with frequency", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 433, + 504, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 137, + 445 + ], + "score": 0.88, + "content": "k = 3 5", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 137, + 433, + 504, + 447 + ], + "score": 1.0, + "content": ". In Figure 4, we also plot the activations of MLP neuron 0 for every combination of inputs.", + "type": "text", + "cross_page": true + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 288, + 458 + ], + "score": 1.0, + "content": "The activations are periodic with frequency", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 288, + 446, + 322, + 456 + ], + "score": 0.9, + "content": "k = 4 2", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 322, + 446, + 505, + 458 + ], + "score": 1.0, + "content": ". 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As discussed in", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 175, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 216, + 187 + ], + "score": 1.0, + "content": "Section 4.1, the sparsity of", + "type": "text" + }, + { + "bbox": [ + 216, + 175, + 233, + 186 + ], + "score": 0.9, + "content": "W _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 176, + 505, + 187 + ], + "score": 1.0, + "content": "in the Fourier basis is evidence that the network is operating in this", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 186, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 198 + ], + "score": 1.0, + "content": "basis. 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(Right) Norm of Fourier components of the neuron-logit", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 208, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 126, + 220 + ], + "score": 1.0, + "content": "map", + "type": "text" + }, + { + "bbox": [ + 126, + 208, + 142, + 219 + ], + "score": 0.88, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 208, + 505, + 220 + ], + "score": 1.0, + "content": ". A Fourier transform is taken over the logit axis, and then the norm is taken over the neuron", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 218, + 464, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 242, + 232 + ], + "score": 1.0, + "content": "axis. 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(Center) The activations of MLP neuron 0 given inputs", + "type": "text" + }, + { + "bbox": [ + 351, + 377, + 366, + 388 + ], + "score": 0.88, + "content": "a , b", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 376, + 505, + 389 + ], + "score": 1.0, + "content": ". Both the attention scores and the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "neuron activations are periodic (Section 4.1). 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In Figure 4, we also plot the activations of MLP neuron 0 for every combination of inputs.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 288, + 458 + ], + "score": 1.0, + "content": "The activations are periodic with frequency", + "type": "text" + }, + { + "bbox": [ + 288, + 446, + 322, + 456 + ], + "score": 0.9, + "content": "k = 4 2", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 446, + 505, + 458 + ], + "score": 1.0, + "content": ". We see similar patterns for other attention", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 456, + 273, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 273, + 469 + ], + "score": 1.0, + "content": "heads and MLP neurons (Appendix C.1).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 473, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "Periodicity in logits. Finally, the logits are also periodic. In Figure 4, we represent the logits in the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 295, + 497 + ], + "score": 1.0, + "content": "2D Fourier basis over the inputs, then take the", + "type": "text" + }, + { + "bbox": [ + 296, + 485, + 306, + 496 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "-norm over the output dimension. There are only", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 496, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 507 + ], + "score": 1.0, + "content": "twenty components with significant norm, corresponding to the products of sines and cosines for the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 506, + 398, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 188, + 519 + ], + "score": 1.0, + "content": "five key frequencies", + "type": "text" + }, + { + "bbox": [ + 189, + 508, + 201, + 518 + ], + "score": 0.85, + "content": "w _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 506, + 296, + 519 + ], + "score": 1.0, + "content": ". These show up as five", + "type": "text" + }, + { + "bbox": [ + 296, + 507, + 320, + 517 + ], + "score": 0.9, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 506, + 398, + 519 + ], + "score": 1.0, + "content": "blocks in Figure 4.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "title", + "bbox": [ + 107, + 531, + 379, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 531, + 380, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 380, + 544 + ], + "score": 1.0, + "content": "4.2 MECHANISTIC EVIDENCE: COMPOSING MODEL WEIGHTS", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 551, + 504, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "score": 1.0, + "content": "We now demonstrate that the model implements the trigonometric identity (1) as follows: the func-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 129, + 576 + ], + "score": 1.0, + "content": "tions", + "type": "text" + }, + { + "bbox": [ + 129, + 563, + 192, + 575 + ], + "score": 0.85, + "content": "\\cos { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 562, + 198, + 576 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 198, + 563, + 262, + 575 + ], + "score": 0.88, + "content": "\\sin { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 562, + 506, + 576 + ], + "score": 1.0, + "content": "are linearly represented in the MLP activations, and the un-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 573, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 384, + 587 + ], + "score": 1.0, + "content": "embed matrix reads these linear directions and multiplies them by cos", + "type": "text" + }, + { + "bbox": [ + 385, + 574, + 409, + 586 + ], + "score": 0.61, + "content": "( w _ { k } c )", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 573, + 412, + 587 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 413, + 574, + 452, + 586 + ], + "score": 0.74, + "content": "\\sin { ( w _ { k } c ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 573, + 505, + 587 + ], + "score": 1.0, + "content": "respectively.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 590, + 502, + 613 + ], + "lines": [ + { + "bbox": [ + 106, + 589, + 504, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 298, + 604 + ], + "score": 1.0, + "content": "We will do this in two steps. First, we show that", + "type": "text" + }, + { + "bbox": [ + 299, + 591, + 315, + 602 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 589, + 504, + 604 + ], + "score": 1.0, + "content": "(the matrix mapping MLP activations to logits)", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 601, + 351, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 351, + 615 + ], + "score": 1.0, + "content": "is (approximately) rank 10 and can be well approximated as:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 618, + 417, + 640 + ], + "lines": [ + { + "bbox": [ + 194, + 618, + 417, + 640 + ], + "spans": [ + { + "bbox": [ + 194, + 618, + 417, + 640 + ], + "score": 0.91, + "content": "\\boldsymbol { W _ { L } } = \\sum _ { k \\in \\left\\{ 1 4 , 3 5 , 4 1 , 4 2 , 5 2 \\right\\} } \\cos \\left( \\boldsymbol { w _ { k } } \\right) \\boldsymbol { u _ { k } ^ { T } } + \\sin \\left( \\boldsymbol { w _ { k } } \\right) \\boldsymbol { v _ { k } ^ { T } }", + "type": "interline_equation", + "image_path": "36aeb17aec3a0a165ad912adfe4f3623276afc161390b745310b23f5d1d4d102.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 194, + 618, + 417, + 640 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 644, + 504, + 668 + ], + "lines": [ + { + "bbox": [ + 104, + 641, + 504, + 660 + ], + "spans": [ + { + "bbox": [ + 104, + 641, + 145, + 660 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 145, + 644, + 202, + 656 + ], + "score": 0.91, + "content": "u _ { k } , v _ { k } \\in \\mathbb { R } ^ { 5 1 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 641, + 233, + 660 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 234, + 644, + 341, + 657 + ], + "score": 0.9, + "content": "\\cos \\left( w _ { k } \\right) , \\sin \\left( w _ { k } \\right) \\in \\mathbb { R } ^ { 1 1 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 641, + 463, + 660 + ], + "score": 1.0, + "content": "are vectors whose cth entry is", + "type": "text" + }, + { + "bbox": [ + 464, + 645, + 504, + 657 + ], + "score": 0.9, + "content": "\\cos { \\left( w _ { k } c \\right) }", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 655, + 414, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 123, + 669 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 657, + 162, + 668 + ], + "score": 0.92, + "content": "\\sin { ( w _ { k } c ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 655, + 384, + 669 + ], + "score": 1.0, + "content": ". Second, note that our model implements the logits for", + "type": "text" + }, + { + "bbox": [ + 384, + 657, + 399, + 668 + ], + "score": 0.89, + "content": "a , b", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 655, + 414, + 669 + ], + "score": 1.0, + "content": "as:", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 672, + 479, + 693 + ], + "lines": [ + { + "bbox": [ + 131, + 672, + 479, + 693 + ], + "spans": [ + { + "bbox": [ + 131, + 672, + 479, + 693 + ], + "score": 0.91, + "content": "\\mathrm { L o g i t s } ( a , b ) = W _ { L } \\mathrm { M L P } ( a , b ) \\approx \\sum _ { k } \\cos \\left( w _ { k } \\right) u _ { k } ^ { T } \\mathrm { M L P } ( a , b ) + \\sin \\left( w _ { k } \\right) v _ { k } ^ { T } \\mathrm { M L P } ( a , b )", + "type": "interline_equation", + "image_path": "54fe47fb0fd113e6323d3a8112093b7899f1b64a07a5058f475c331f2ccf34f5.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 131, + 672, + 479, + 693 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 697, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 697, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 255, + 712 + ], + "score": 1.0, + "content": "We check empirically that the terms", + "type": "text" + }, + { + "bbox": [ + 255, + 698, + 310, + 711 + ], + "score": 0.91, + "content": "u _ { k } ^ { T } \\mathbf { M } \\mathbf { L P } ( a , b )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 697, + 330, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 330, + 698, + 384, + 711 + ], + "score": 0.92, + "content": "v _ { k } ^ { T } \\mathbf { M L P } ( a , b )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 697, + 506, + 712 + ], + "score": 1.0, + "content": "are approximate multiples of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 107, + 709, + 172, + 722 + ], + "score": 0.9, + "content": "\\cos { ( w _ { k } ( a + \\bar { b } ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 709, + 191, + 723 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 192, + 710, + 254, + 722 + ], + "score": 0.82, + "content": "\\sin \\left( w _ { k } ( a + b ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 709, + 258, + 723 + ], + "score": 1.0, + "content": ")", + "type": "text" + }, + { + "bbox": [ + 258, + 710, + 291, + 721 + ], + "score": 0.71, + "content": "( > 9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "of variance explained). Thus the network computes", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 720, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 504, + 732 + ], + "score": 1.0, + "content": "trigonometric functions in the MLP and reads them off as claimed. 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(Center) The activations of MLP neuron 0 given inputs", + "type": "text" + }, + { + "bbox": [ + 351, + 377, + 366, + 388 + ], + "score": 0.88, + "content": "a , b", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 376, + 505, + 389 + ], + "score": 1.0, + "content": ". Both the attention scores and the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "neuron activations are periodic (Section 4.1). (Right) The norm of the Fourier components of the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 315, + 411 + ], + "score": 1.0, + "content": "logits (2D Fourier transform is taken over the inputs", + "type": "text" + }, + { + "bbox": [ + 315, + 399, + 330, + 410 + ], + "score": 0.88, + "content": "a , b", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 398, + 505, + 411 + ], + "score": 1.0, + "content": ", and then norm is taken over the logit axis).", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 408, + 467, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 467, + 422 + ], + "score": 1.0, + "content": "There are 20 significant components corresponding to the 5 key frequencies (Section 4.1).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14 + } + ], + "index": 12.0 + }, + { + "type": "text", + "bbox": [ + 106, + 434, + 505, + 468 + ], + "lines": [], + "index": 18, + "bbox_fs": [ + 105, + 433, + 505, + 469 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 473, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "Periodicity in logits. Finally, the logits are also periodic. In Figure 4, we represent the logits in the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 295, + 497 + ], + "score": 1.0, + "content": "2D Fourier basis over the inputs, then take the", + "type": "text" + }, + { + "bbox": [ + 296, + 485, + 306, + 496 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "-norm over the output dimension. 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These show up as five", + "type": "text" + }, + { + "bbox": [ + 296, + 507, + 320, + 517 + ], + "score": 0.9, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 506, + 398, + 519 + ], + "score": 1.0, + "content": "blocks in Figure 4.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 473, + 505, + 519 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 531, + 379, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 531, + 380, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 380, + 544 + ], + "score": 1.0, + "content": "4.2 MECHANISTIC EVIDENCE: COMPOSING MODEL WEIGHTS", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 551, + 504, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "score": 1.0, + "content": "We now demonstrate that the model implements the trigonometric identity (1) as follows: the func-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 129, + 576 + ], + "score": 1.0, + "content": "tions", + "type": "text" + }, + { + "bbox": [ + 129, + 563, + 192, + 575 + ], + "score": 0.85, + "content": "\\cos { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 562, + 198, + 576 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 198, + 563, + 262, + 575 + ], + "score": 0.88, + "content": "\\sin { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 562, + 506, + 576 + ], + "score": 1.0, + "content": "are linearly represented in the MLP activations, and the un-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 573, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 384, + 587 + ], + "score": 1.0, + "content": "embed matrix reads these linear directions and multiplies them by cos", + "type": "text" + }, + { + "bbox": [ + 385, + 574, + 409, + 586 + ], + "score": 0.61, + "content": "( w _ { k } c )", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 573, + 412, + 587 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 413, + 574, + 452, + 586 + ], + "score": 0.74, + "content": "\\sin { ( w _ { k } c ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 573, + 505, + 587 + ], + "score": 1.0, + "content": "respectively.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 551, + 506, + 587 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 590, + 502, + 613 + ], + "lines": [ + { + "bbox": [ + 106, + 589, + 504, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 298, + 604 + ], + "score": 1.0, + "content": "We will do this in two steps. First, we show that", + "type": "text" + }, + { + "bbox": [ + 299, + 591, + 315, + 602 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 589, + 504, + 604 + ], + "score": 1.0, + "content": "(the matrix mapping MLP activations to logits)", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 601, + 351, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 351, + 615 + ], + "score": 1.0, + "content": "is (approximately) rank 10 and can be well approximated as:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 589, + 504, + 615 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 618, + 417, + 640 + ], + "lines": [ + { + "bbox": [ + 194, + 618, + 417, + 640 + ], + "spans": [ + { + "bbox": [ + 194, + 618, + 417, + 640 + ], + "score": 0.91, + "content": "\\boldsymbol { W _ { L } } = \\sum _ { k \\in \\left\\{ 1 4 , 3 5 , 4 1 , 4 2 , 5 2 \\right\\} } \\cos \\left( \\boldsymbol { w _ { k } } \\right) \\boldsymbol { u _ { k } ^ { T } } + \\sin \\left( \\boldsymbol { w _ { k } } \\right) \\boldsymbol { v _ { k } ^ { T } }", + "type": "interline_equation", + "image_path": "36aeb17aec3a0a165ad912adfe4f3623276afc161390b745310b23f5d1d4d102.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 194, + 618, + 417, + 640 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 644, + 504, + 668 + ], + "lines": [ + { + "bbox": [ + 104, + 641, + 504, + 660 + ], + "spans": [ + { + "bbox": [ + 104, + 641, + 145, + 660 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 145, + 644, + 202, + 656 + ], + "score": 0.91, + "content": "u _ { k } , v _ { k } \\in \\mathbb { R } ^ { 5 1 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 641, + 233, + 660 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 234, + 644, + 341, + 657 + ], + "score": 0.9, + "content": "\\cos \\left( w _ { k } \\right) , \\sin \\left( w _ { k } \\right) \\in \\mathbb { R } ^ { 1 1 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 641, + 463, + 660 + ], + "score": 1.0, + "content": "are vectors whose cth entry is", + "type": "text" + }, + { + "bbox": [ + 464, + 645, + 504, + 657 + ], + "score": 0.9, + "content": "\\cos { \\left( w _ { k } c \\right) }", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 655, + 414, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 123, + 669 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 657, + 162, + 668 + ], + "score": 0.92, + "content": "\\sin { ( w _ { k } c ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 655, + 384, + 669 + ], + "score": 1.0, + "content": ". Second, note that our model implements the logits for", + "type": "text" + }, + { + "bbox": [ + 384, + 657, + 399, + 668 + ], + "score": 0.89, + "content": "a , b", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 655, + 414, + 669 + ], + "score": 1.0, + "content": "as:", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 641, + 504, + 669 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 672, + 479, + 693 + ], + "lines": [ + { + "bbox": [ + 131, + 672, + 479, + 693 + ], + "spans": [ + { + "bbox": [ + 131, + 672, + 479, + 693 + ], + "score": 0.91, + "content": "\\mathrm { L o g i t s } ( a , b ) = W _ { L } \\mathrm { M L P } ( a , b ) \\approx \\sum _ { k } \\cos \\left( w _ { k } \\right) u _ { k } ^ { T } \\mathrm { M L P } ( a , b ) + \\sin \\left( w _ { k } \\right) v _ { k } ^ { T } \\mathrm { M L P } ( a , b )", + "type": "interline_equation", + "image_path": "54fe47fb0fd113e6323d3a8112093b7899f1b64a07a5058f475c331f2ccf34f5.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 131, + 672, + 479, + 693 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 697, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 697, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 255, + 712 + ], + "score": 1.0, + "content": "We check empirically that the terms", + "type": "text" + }, + { + "bbox": [ + 255, + 698, + 310, + 711 + ], + "score": 0.91, + "content": "u _ { k } ^ { T } \\mathbf { M } \\mathbf { L P } ( a , b )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 697, + 330, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 330, + 698, + 384, + 711 + ], + "score": 0.92, + "content": "v _ { k } ^ { T } \\mathbf { M L P } ( a , b )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 697, + 506, + 712 + ], + "score": 1.0, + "content": "are approximate multiples of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 107, + 709, + 172, + 722 + ], + "score": 0.9, + "content": "\\cos { ( w _ { k } ( a + \\bar { b } ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 709, + 191, + 723 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 192, + 710, + 254, + 722 + ], + "score": 0.82, + "content": "\\sin \\left( w _ { k } ( a + b ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 709, + 258, + 723 + ], + "score": 1.0, + "content": ")", + "type": "text" + }, + { + "bbox": [ + 258, + 710, + 291, + 721 + ], + "score": 0.71, + "content": "( > 9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "of variance explained). Thus the network computes", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 720, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 504, + 732 + ], + "score": 1.0, + "content": "trigonometric functions in the MLP and reads them off as claimed. As a sanity check, we confirm", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 697, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 107, + 54, + 519, + 169 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 107, + 54, + 519, + 169 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 54, + 519, + 169 + ], + "spans": [ + { + "bbox": [ + 107, + 54, + 519, + 169 + ], + "score": 0.98, + "html": "
WL ComponentFourier components of uTMLP(a,b) or uTMLP(a, b) FVE
cos (W14C)44.6cos(w14a) cos(w14b)- 43.6 sin(w14a) sin(w14b)~ 44.1 cos (w14(a + b))93.2%
sin (w14c)44.1 sin(w14a) cos(w14b) + 44.1 cos(w14a) sin(w14b) ~ 44.1 sin (w14(a + b))93.5%
cos (W35C)40.7 cos(w35a) cos(w35b)- 43.6 sin(w35a) sin(w35b) ~ 42.2 cos (w35(a + b))96.8%
sin (w35C)41.8 sin(w35a) cos(w35b) + 41.8 cos(w35a) sin(w35b) ~ 41.8 sin (w35(a + b))96.5%
cos (w41c)44.8 cos(w41a) cos(w41b) - 44.8 sin(w41a) sin(w41b) ~ 44.8 cos (w41(a + b))97.0%
sin (w41c)44.5 sin(w41a) cos(w41b) + 44.5 cos(w41a) sin(w41b) ~ 44.5 sin (w41(a + b))97.0%
cos (W42C)64.6 cos(w42a) cos(w42b) - 68.5 sin(w42a) sin(w42b) ~ 66.6 cos (w42(a + b))96.4%
sin (w42c)67.8 sin(w42a) cos(w42b) + 67.8 cos(w42a) sin(w42b) ~ 67.8 sin (w42(a + b))96.4%
cos (W52C)60.5 cos(w52a) cos(w52b) - 65.5 sin(w52a) sin(w52b) ~ 63.0cos (w52(a + b))97.4%
sin (w52C)64.5 sin(w52a) cos(w52b) + 64.5 cos(w52a) sin(w52b) ~ 64.5 sin (w52(a + b))98.2%
", + "type": "table", + "image_path": "d3f9501e4b42b566f9605d2ecb00c692ef82094ad5322735dfcabf79f73d2323.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 54, + 519, + 92.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 92.33333333333334, + 519, + 130.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 130.66666666666669, + 519, + 169.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 176, + 506, + 232 + ], + "lines": [ + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 245, + 189 + ], + "score": 1.0, + "content": "Table 1: For each of the directions", + "type": "text" + }, + { + "bbox": [ + 245, + 178, + 257, + 187 + ], + "score": 0.86, + "content": "u _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 176, + 268, + 189 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 268, + 178, + 280, + 187 + ], + "score": 0.85, + "content": "v _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 176, + 367, + 189 + ], + "score": 1.0, + "content": "(corresponding to the", + "type": "text" + }, + { + "bbox": [ + 368, + 176, + 402, + 188 + ], + "score": 0.9, + "content": "\\cos ( w _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 176, + 420, + 189 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 420, + 176, + 453, + 188 + ], + "score": 0.89, + "content": "\\sin ( w _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "components", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "respectively) in the unembedding matrix, we take the dot product of the MLP activations with that", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "direction, then perform a Fourier transform (middle column; only two largest coefficients shown).", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "We then compute the fraction of variance explained (FVE) if we replace the projection with a single", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 220, + 498, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 192, + 233 + ], + "score": 1.0, + "content": "term proportional to", + "type": "text" + }, + { + "bbox": [ + 192, + 220, + 254, + 232 + ], + "score": 0.89, + "content": "\\cos \\left( w _ { k } ( a + b ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 220, + 267, + 233 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 267, + 220, + 329, + 232 + ], + "score": 0.92, + "content": "\\sin \\left( w _ { k } ( a + b ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 220, + 498, + 233 + ], + "score": 1.0, + "content": ", and find that it is consistently close to 1.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 250, + 504, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 248, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 384, + 263 + ], + "score": 1.0, + "content": "that the logits are indeed well-approximated by terms of the form", + "type": "text" + }, + { + "bbox": [ + 385, + 250, + 466, + 262 + ], + "score": 0.78, + "content": "\\cos { \\left( w _ { k } ( a + b - c ) \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 250, + 492, + 261 + ], + "score": 0.81, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 248, + 506, + 263 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 261, + 189, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 189, + 273 + ], + "score": 1.0, + "content": "variance explained).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 106, + 277, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 106, + 277, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 123, + 289 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 277, + 231, + 292 + ], + "score": 1.0, + "content": "is well approximated by", + "type": "text" + }, + { + "bbox": [ + 231, + 278, + 272, + 290 + ], + "score": 0.8, + "content": "\\cos { \\left( w _ { k } c \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 277, + 293, + 292 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 293, + 278, + 332, + 290 + ], + "score": 0.9, + "content": "\\sin { ( w _ { k } c ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 277, + 506, + 292 + ], + "score": 1.0, + "content": ". We perform a discrete Fourier transform", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 289, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 215, + 303 + ], + "score": 1.0, + "content": "(DFT) on the logit axis of", + "type": "text" + }, + { + "bbox": [ + 215, + 290, + 232, + 300 + ], + "score": 0.9, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 289, + 352, + 303 + ], + "score": 1.0, + "content": "and look at the 10 directions", + "type": "text" + }, + { + "bbox": [ + 353, + 291, + 379, + 300 + ], + "score": 0.88, + "content": "u _ { k } , v _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 289, + 451, + 303 + ], + "score": 1.0, + "content": "corresponding to", + "type": "text" + }, + { + "bbox": [ + 451, + 289, + 486, + 301 + ], + "score": 0.89, + "content": "\\sin \\left( w _ { k } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 289, + 505, + 303 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 298, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 142, + 312 + ], + "score": 0.88, + "content": "\\cos \\left( \\boldsymbol { w } _ { k } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 298, + 238, + 316 + ], + "score": 1.0, + "content": ". When we approximate", + "type": "text" + }, + { + "bbox": [ + 239, + 300, + 255, + 312 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 298, + 275, + 316 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 276, + 300, + 462, + 313 + ], + "score": 0.81, + "content": "\\begin{array} { r } { \\sum _ { k \\in \\{ 1 4 , 3 5 , 4 1 , 4 2 , 5 2 \\} } \\cos \\left( w _ { k } \\right) u _ { k } ^ { T } + \\sin \\left( w _ { k } \\right) v _ { k } ^ { T } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 298, + 506, + 316 + ], + "score": 1.0, + "content": ", the resid-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 312, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 259, + 327 + ], + "score": 1.0, + "content": "ual has Frobenius norm that is under", + "type": "text" + }, + { + "bbox": [ + 259, + 313, + 287, + 324 + ], + "score": 0.88, + "content": "0 . 5 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 312, + 351, + 327 + ], + "score": 1.0, + "content": "of the norm of", + "type": "text" + }, + { + "bbox": [ + 351, + 314, + 367, + 324 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 312, + 441, + 327 + ], + "score": 1.0, + "content": ". This shows that", + "type": "text" + }, + { + "bbox": [ + 441, + 313, + 457, + 324 + ], + "score": 0.88, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 312, + 505, + 327 + ], + "score": 1.0, + "content": "is well ap-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 324, + 504, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 312, + 336 + ], + "score": 1.0, + "content": "proximated by the 10 directions corresponding to", + "type": "text" + }, + { + "bbox": [ + 313, + 324, + 348, + 336 + ], + "score": 0.89, + "content": "\\cos \\left( w _ { k } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 324, + 368, + 336 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 369, + 325, + 403, + 336 + ], + "score": 0.91, + "content": "\\sin \\left( w _ { k } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 324, + 504, + 336 + ], + "score": 1.0, + "content": "for each of the five key", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 334, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 348 + ], + "score": 1.0, + "content": "frequencies. We also plot the norms of each direction in Figure 3, and find that no Fourier compo-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 346, + 330, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 330, + 359 + ], + "score": 1.0, + "content": "nent outside the 5 key frequencies has significant norm.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 363, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 503, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 354, + 376 + ], + "score": 1.0, + "content": "The unembedding matrix “reads off” terms of the form", + "type": "text" + }, + { + "bbox": [ + 354, + 363, + 417, + 375 + ], + "score": 0.83, + "content": "\\cos \\left( w _ { k } ( a + b ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 362, + 439, + 376 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 439, + 363, + 503, + 375 + ], + "score": 0.9, + "content": "\\sin { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "score": 1.0, + "content": "from the MLP neurons. Next, we take the dot product of the MLP activations with each of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 164, + 397 + ], + "score": 1.0, + "content": "the directions", + "type": "text" + }, + { + "bbox": [ + 165, + 387, + 191, + 396 + ], + "score": 0.88, + "content": "u _ { k } , v _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 384, + 208, + 397 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 208, + 385, + 308, + 397 + ], + "score": 0.91, + "content": "k \\in \\{ 1 4 , 3 5 , 4 1 , 4 2 , 5 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 384, + 506, + 397 + ], + "score": 1.0, + "content": ". Table 1 displays the results: the dot products", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 395, + 467, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 162, + 408 + ], + "score": 0.93, + "content": "u _ { k } ^ { T } \\mathbf { M } \\mathbf { L P } ( a , b )", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 395, + 180, + 409 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 180, + 396, + 235, + 408 + ], + "score": 0.91, + "content": "v _ { k } ^ { T } \\mathbf { M L P } ( a , b )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 395, + 467, + 409 + ], + "score": 1.0, + "content": "are well approximated by a multiple of terms of the form", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "interline_equation", + "bbox": [ + 167, + 409, + 421, + 437 + ], + "lines": [ + { + "bbox": [ + 167, + 409, + 421, + 437 + ], + "spans": [ + { + "bbox": [ + 167, + 409, + 421, + 437 + ], + "score": 0.83, + "content": "\\begin{array} { r } { \\cos \\left( { { w } _ { k } } ( a + b ) \\right) = \\cos \\left( { { w } _ { k } } a \\right) \\cos \\left( { { w } _ { k } } b \\right) - \\sin \\left( { { w } _ { k } } a \\right) \\sin \\left( { { w } _ { k } } b \\right) , } \\\\ { \\sin \\left( { { w } _ { k } } ( a + b ) \\right) = \\sin \\left( { { w } _ { k } } a \\right) \\cos \\left( { { w } _ { k } } b \\right) + \\cos \\left( { { w } _ { k } } a \\right) \\sin \\left( { { w } _ { k } } b \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "7e3aee2a04550a055d9a7ea68aa8cbc62df2107387237f3040ddf6c466849709.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 167, + 409, + 421, + 418.3333333333333 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 167, + 418.3333333333333, + 421, + 427.66666666666663 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 167, + 427.66666666666663, + 421, + 436.99999999999994 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 438, + 504, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 240, + 452 + ], + "score": 1.0, + "content": "That is, for each key frequency", + "type": "text" + }, + { + "bbox": [ + 241, + 439, + 265, + 449 + ], + "score": 0.43, + "content": "k , \\ u _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 437, + 285, + 452 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 286, + 441, + 297, + 450 + ], + "score": 0.86, + "content": "v _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 437, + 506, + 452 + ], + "score": 1.0, + "content": "are linear directions in the space of MLP neuron", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 450, + 360, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 209, + 462 + ], + "score": 1.0, + "content": "activations that represent", + "type": "text" + }, + { + "bbox": [ + 209, + 450, + 273, + 462 + ], + "score": 0.89, + "content": "\\cos { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 450, + 292, + 462 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 292, + 450, + 355, + 462 + ], + "score": 0.9, + "content": "\\sin { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 450, + 360, + 462 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 466, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 330, + 479 + ], + "score": 1.0, + "content": "Logits are well approximated by a weighted sum of", + "type": "text" + }, + { + "bbox": [ + 330, + 466, + 416, + 478 + ], + "score": 0.86, + "content": "\\cos { \\left( w _ { k } ( a + b - c ) \\right) } \\mathbf { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 466, + 505, + 479 + ], + "score": 1.0, + "content": ". We approximate the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 477, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 104, + 477, + 201, + 490 + ], + "score": 1.0, + "content": "output logits as the sum", + "type": "text" + }, + { + "bbox": [ + 201, + 477, + 303, + 490 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\sum _ { k } \\alpha _ { k } \\cos ( w _ { k } ( a + b - c ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 477, + 318, + 490 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 318, + 478, + 415, + 489 + ], + "score": 0.88, + "content": "k \\in \\{ 1 4 , 3 5 , 4 1 , 4 2 , 5 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 477, + 506, + 490 + ], + "score": 1.0, + "content": "and fit the coefficients", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 119, + 500 + ], + "score": 0.84, + "content": "\\alpha _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 488, + 356, + 501 + ], + "score": 1.0, + "content": "via ordinary least squares. This approximation explains", + "type": "text" + }, + { + "bbox": [ + 357, + 489, + 376, + 499 + ], + "score": 0.86, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "of the variance in the original", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 499, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 306, + 512 + ], + "score": 1.0, + "content": "logits. This is surprising—the output logits are a", + "type": "text" + }, + { + "bbox": [ + 307, + 500, + 369, + 510 + ], + "score": 0.85, + "content": "1 1 3 \\cdot 1 1 3 \\cdot 1 1 3", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 499, + 505, + 512 + ], + "score": 1.0, + "content": "dimensional vector, but are well-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 104, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "approximated with just the 5 directions predicted by our interpretation. If we evaluate test loss using", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 520, + 488, + 533 + ], + "spans": [ + { + "bbox": [ + 104, + 520, + 389, + 533 + ], + "score": 1.0, + "content": "this logit approximation, we actually see an improvement in loss, from", + "type": "text" + }, + { + "bbox": [ + 389, + 521, + 431, + 532 + ], + "score": 0.89, + "content": "2 . 4 \\cdot 1 0 ^ { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 520, + 442, + 533 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 443, + 521, + 484, + 532 + ], + "score": 0.89, + "content": "4 . 7 \\cdot 1 0 ^ { - 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 520, + 488, + 533 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 105, + 538, + 504, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "Taken together, these results confirm that the model computes sums of terms of the form", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 548, + 426, + 563 + ], + "spans": [ + { + "bbox": [ + 107, + 549, + 423, + 561 + ], + "score": 0.84, + "content": "\\cos { \\left( w _ { k } ( \\bar { a } + b - c ) \\right) } = \\cos { \\left( w _ { k } ( a + b ) \\right) } \\cos { \\left( w _ { k } c \\right) } + \\sin { \\left( w _ { k } ( \\bar { a + } b ) \\right) } \\sin { \\left( w _ { k } c \\right) } .", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 548, + 426, + 563 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 107, + 573, + 425, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 573, + 426, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 426, + 586 + ], + "score": 1.0, + "content": "4.3 ZOOMING IN: APPROXIMATING NEURONS WITH SINES AND COSINES", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 593, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 452, + 606 + ], + "score": 1.0, + "content": "In the previous section, we showed how the model computes its final logits by using", + "type": "text" + }, + { + "bbox": [ + 452, + 594, + 468, + 605 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "to “read", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "off” trigonometric identities represented in the MLP neurons. We now examine the attention heads", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 617, + 504, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 504, + 627 + ], + "score": 1.0, + "content": "and MLP neurons to understand how the identities come to be represented at the MLP layer. In", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "Appendix C.1.2, we show that two of the attention heads approximately compute degree-2 poly-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "nomials of sines and cosines of a particular frequency (and the other two are used to increase the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "magnitude of the input embeddings in the residual stream). Here, we show that most neurons are", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "also well-approximated by degree-2 polynomials, and the map from neurons to logits is localized by", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 670, + 151, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 151, + 685 + ], + "score": 1.0, + "content": "frequency.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 107, + 688, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Most MLP neurons approximately compute a degree-2 polynomial of a single frequency. We", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "next try to approximate the activations of each MLP neuron by a degree-2 polynomial of one of the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 397, + 722 + ], + "score": 1.0, + "content": "5 key frequencies. As shown in Figure 5, out of 512 total neurons, 433", + "type": "text" + }, + { + "bbox": [ + 398, + 710, + 428, + 721 + ], + "score": 0.8, + "content": "( 8 4 . 6 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 709, + 473, + 722 + ], + "score": 1.0, + "content": ") have over", + "type": "text" + }, + { + "bbox": [ + 473, + 709, + 493, + 720 + ], + "score": 0.88, + "content": "8 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 302, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 302, + 734 + ], + "score": 1.0, + "content": "their variance explained with a single frequency.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 107, + 54, + 519, + 169 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 107, + 54, + 519, + 169 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 54, + 519, + 169 + ], + "spans": [ + { + "bbox": [ + 107, + 54, + 519, + 169 + ], + "score": 0.98, + "html": "
WL ComponentFourier components of uTMLP(a,b) or uTMLP(a, b) FVE
cos (W14C)44.6cos(w14a) cos(w14b)- 43.6 sin(w14a) sin(w14b)~ 44.1 cos (w14(a + b))93.2%
sin (w14c)44.1 sin(w14a) cos(w14b) + 44.1 cos(w14a) sin(w14b) ~ 44.1 sin (w14(a + b))93.5%
cos (W35C)40.7 cos(w35a) cos(w35b)- 43.6 sin(w35a) sin(w35b) ~ 42.2 cos (w35(a + b))96.8%
sin (w35C)41.8 sin(w35a) cos(w35b) + 41.8 cos(w35a) sin(w35b) ~ 41.8 sin (w35(a + b))96.5%
cos (w41c)44.8 cos(w41a) cos(w41b) - 44.8 sin(w41a) sin(w41b) ~ 44.8 cos (w41(a + b))97.0%
sin (w41c)44.5 sin(w41a) cos(w41b) + 44.5 cos(w41a) sin(w41b) ~ 44.5 sin (w41(a + b))97.0%
cos (W42C)64.6 cos(w42a) cos(w42b) - 68.5 sin(w42a) sin(w42b) ~ 66.6 cos (w42(a + b))96.4%
sin (w42c)67.8 sin(w42a) cos(w42b) + 67.8 cos(w42a) sin(w42b) ~ 67.8 sin (w42(a + b))96.4%
cos (W52C)60.5 cos(w52a) cos(w52b) - 65.5 sin(w52a) sin(w52b) ~ 63.0cos (w52(a + b))97.4%
sin (w52C)64.5 sin(w52a) cos(w52b) + 64.5 cos(w52a) sin(w52b) ~ 64.5 sin (w52(a + b))98.2%
", + "type": "table", + "image_path": "d3f9501e4b42b566f9605d2ecb00c692ef82094ad5322735dfcabf79f73d2323.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 54, + 519, + 92.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 92.33333333333334, + 519, + 130.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 130.66666666666669, + 519, + 169.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 176, + 506, + 232 + ], + "lines": [ + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 245, + 189 + ], + "score": 1.0, + "content": "Table 1: For each of the directions", + "type": "text" + }, + { + "bbox": [ + 245, + 178, + 257, + 187 + ], + "score": 0.86, + "content": "u _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 176, + 268, + 189 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 268, + 178, + 280, + 187 + ], + "score": 0.85, + "content": "v _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 176, + 367, + 189 + ], + "score": 1.0, + "content": "(corresponding to the", + "type": "text" + }, + { + "bbox": [ + 368, + 176, + 402, + 188 + ], + "score": 0.9, + "content": "\\cos ( w _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 176, + 420, + 189 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 420, + 176, + 453, + 188 + ], + "score": 0.89, + "content": "\\sin ( w _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "components", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "respectively) in the unembedding matrix, we take the dot product of the MLP activations with that", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "direction, then perform a Fourier transform (middle column; only two largest coefficients shown).", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "We then compute the fraction of variance explained (FVE) if we replace the projection with a single", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 220, + 498, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 192, + 233 + ], + "score": 1.0, + "content": "term proportional to", + "type": "text" + }, + { + "bbox": [ + 192, + 220, + 254, + 232 + ], + "score": 0.89, + "content": "\\cos \\left( w _ { k } ( a + b ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 220, + 267, + 233 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 267, + 220, + 329, + 232 + ], + "score": 0.92, + "content": "\\sin \\left( w _ { k } ( a + b ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 220, + 498, + 233 + ], + "score": 1.0, + "content": ", and find that it is consistently close to 1.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 176, + 505, + 233 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 250, + 504, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 248, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 384, + 263 + ], + "score": 1.0, + "content": "that the logits are indeed well-approximated by terms of the form", + "type": "text" + }, + { + "bbox": [ + 385, + 250, + 466, + 262 + ], + "score": 0.78, + "content": "\\cos { \\left( w _ { k } ( a + b - c ) \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 250, + 492, + 261 + ], + "score": 0.81, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 248, + 506, + 263 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 261, + 189, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 189, + 273 + ], + "score": 1.0, + "content": "variance explained).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 248, + 506, + 273 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 277, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 106, + 277, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 123, + 289 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 277, + 231, + 292 + ], + "score": 1.0, + "content": "is well approximated by", + "type": "text" + }, + { + "bbox": [ + 231, + 278, + 272, + 290 + ], + "score": 0.8, + "content": "\\cos { \\left( w _ { k } c \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 277, + 293, + 292 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 293, + 278, + 332, + 290 + ], + "score": 0.9, + "content": "\\sin { ( w _ { k } c ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 277, + 506, + 292 + ], + "score": 1.0, + "content": ". We perform a discrete Fourier transform", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 289, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 215, + 303 + ], + "score": 1.0, + "content": "(DFT) on the logit axis of", + "type": "text" + }, + { + "bbox": [ + 215, + 290, + 232, + 300 + ], + "score": 0.9, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 289, + 352, + 303 + ], + "score": 1.0, + "content": "and look at the 10 directions", + "type": "text" + }, + { + "bbox": [ + 353, + 291, + 379, + 300 + ], + "score": 0.88, + "content": "u _ { k } , v _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 289, + 451, + 303 + ], + "score": 1.0, + "content": "corresponding to", + "type": "text" + }, + { + "bbox": [ + 451, + 289, + 486, + 301 + ], + "score": 0.89, + "content": "\\sin \\left( w _ { k } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 289, + 505, + 303 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 298, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 142, + 312 + ], + "score": 0.88, + "content": "\\cos \\left( \\boldsymbol { w } _ { k } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 298, + 238, + 316 + ], + "score": 1.0, + "content": ". When we approximate", + "type": "text" + }, + { + "bbox": [ + 239, + 300, + 255, + 312 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 298, + 275, + 316 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 276, + 300, + 462, + 313 + ], + "score": 0.81, + "content": "\\begin{array} { r } { \\sum _ { k \\in \\{ 1 4 , 3 5 , 4 1 , 4 2 , 5 2 \\} } \\cos \\left( w _ { k } \\right) u _ { k } ^ { T } + \\sin \\left( w _ { k } \\right) v _ { k } ^ { T } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 298, + 506, + 316 + ], + "score": 1.0, + "content": ", the resid-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 312, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 259, + 327 + ], + "score": 1.0, + "content": "ual has Frobenius norm that is under", + "type": "text" + }, + { + "bbox": [ + 259, + 313, + 287, + 324 + ], + "score": 0.88, + "content": "0 . 5 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 312, + 351, + 327 + ], + "score": 1.0, + "content": "of the norm of", + "type": "text" + }, + { + "bbox": [ + 351, + 314, + 367, + 324 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 312, + 441, + 327 + ], + "score": 1.0, + "content": ". This shows that", + "type": "text" + }, + { + "bbox": [ + 441, + 313, + 457, + 324 + ], + "score": 0.88, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 312, + 505, + 327 + ], + "score": 1.0, + "content": "is well ap-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 324, + 504, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 312, + 336 + ], + "score": 1.0, + "content": "proximated by the 10 directions corresponding to", + "type": "text" + }, + { + "bbox": [ + 313, + 324, + 348, + 336 + ], + "score": 0.89, + "content": "\\cos \\left( w _ { k } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 324, + 368, + 336 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 369, + 325, + 403, + 336 + ], + "score": 0.91, + "content": "\\sin \\left( w _ { k } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 324, + 504, + 336 + ], + "score": 1.0, + "content": "for each of the five key", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 334, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 348 + ], + "score": 1.0, + "content": "frequencies. We also plot the norms of each direction in Figure 3, and find that no Fourier compo-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 346, + 330, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 330, + 359 + ], + "score": 1.0, + "content": "nent outside the 5 key frequencies has significant norm.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 277, + 506, + 359 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 363, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 503, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 354, + 376 + ], + "score": 1.0, + "content": "The unembedding matrix “reads off” terms of the form", + "type": "text" + }, + { + "bbox": [ + 354, + 363, + 417, + 375 + ], + "score": 0.83, + "content": "\\cos \\left( w _ { k } ( a + b ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 362, + 439, + 376 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 439, + 363, + 503, + 375 + ], + "score": 0.9, + "content": "\\sin { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "score": 1.0, + "content": "from the MLP neurons. Next, we take the dot product of the MLP activations with each of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 164, + 397 + ], + "score": 1.0, + "content": "the directions", + "type": "text" + }, + { + "bbox": [ + 165, + 387, + 191, + 396 + ], + "score": 0.88, + "content": "u _ { k } , v _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 384, + 208, + 397 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 208, + 385, + 308, + 397 + ], + "score": 0.91, + "content": "k \\in \\{ 1 4 , 3 5 , 4 1 , 4 2 , 5 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 384, + 506, + 397 + ], + "score": 1.0, + "content": ". Table 1 displays the results: the dot products", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 395, + 467, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 162, + 408 + ], + "score": 0.93, + "content": "u _ { k } ^ { T } \\mathbf { M } \\mathbf { L P } ( a , b )", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 395, + 180, + 409 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 180, + 396, + 235, + 408 + ], + "score": 0.91, + "content": "v _ { k } ^ { T } \\mathbf { M L P } ( a , b )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 395, + 467, + 409 + ], + "score": 1.0, + "content": "are well approximated by a multiple of terms of the form", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 362, + 506, + 409 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 167, + 409, + 421, + 437 + ], + "lines": [ + { + "bbox": [ + 167, + 409, + 421, + 437 + ], + "spans": [ + { + "bbox": [ + 167, + 409, + 421, + 437 + ], + "score": 0.83, + "content": "\\begin{array} { r } { \\cos \\left( { { w } _ { k } } ( a + b ) \\right) = \\cos \\left( { { w } _ { k } } a \\right) \\cos \\left( { { w } _ { k } } b \\right) - \\sin \\left( { { w } _ { k } } a \\right) \\sin \\left( { { w } _ { k } } b \\right) , } \\\\ { \\sin \\left( { { w } _ { k } } ( a + b ) \\right) = \\sin \\left( { { w } _ { k } } a \\right) \\cos \\left( { { w } _ { k } } b \\right) + \\cos \\left( { { w } _ { k } } a \\right) \\sin \\left( { { w } _ { k } } b \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "7e3aee2a04550a055d9a7ea68aa8cbc62df2107387237f3040ddf6c466849709.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 167, + 409, + 421, + 418.3333333333333 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 167, + 418.3333333333333, + 421, + 427.66666666666663 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 167, + 427.66666666666663, + 421, + 436.99999999999994 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 438, + 504, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 240, + 452 + ], + "score": 1.0, + "content": "That is, for each key frequency", + "type": "text" + }, + { + "bbox": [ + 241, + 439, + 265, + 449 + ], + "score": 0.43, + "content": "k , \\ u _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 437, + 285, + 452 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 286, + 441, + 297, + 450 + ], + "score": 0.86, + "content": "v _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 437, + 506, + 452 + ], + "score": 1.0, + "content": "are linear directions in the space of MLP neuron", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 450, + 360, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 209, + 462 + ], + "score": 1.0, + "content": "activations that represent", + "type": "text" + }, + { + "bbox": [ + 209, + 450, + 273, + 462 + ], + "score": 0.89, + "content": "\\cos { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 450, + 292, + 462 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 292, + 450, + 355, + 462 + ], + "score": 0.9, + "content": "\\sin { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 450, + 360, + 462 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 437, + 506, + 462 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 466, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 330, + 479 + ], + "score": 1.0, + "content": "Logits are well approximated by a weighted sum of", + "type": "text" + }, + { + "bbox": [ + 330, + 466, + 416, + 478 + ], + "score": 0.86, + "content": "\\cos { \\left( w _ { k } ( a + b - c ) \\right) } \\mathbf { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 466, + 505, + 479 + ], + "score": 1.0, + "content": ". We approximate the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 477, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 104, + 477, + 201, + 490 + ], + "score": 1.0, + "content": "output logits as the sum", + "type": "text" + }, + { + "bbox": [ + 201, + 477, + 303, + 490 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\sum _ { k } \\alpha _ { k } \\cos ( w _ { k } ( a + b - c ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 477, + 318, + 490 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 318, + 478, + 415, + 489 + ], + "score": 0.88, + "content": "k \\in \\{ 1 4 , 3 5 , 4 1 , 4 2 , 5 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 477, + 506, + 490 + ], + "score": 1.0, + "content": "and fit the coefficients", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 119, + 500 + ], + "score": 0.84, + "content": "\\alpha _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 488, + 356, + 501 + ], + "score": 1.0, + "content": "via ordinary least squares. This approximation explains", + "type": "text" + }, + { + "bbox": [ + 357, + 489, + 376, + 499 + ], + "score": 0.86, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "of the variance in the original", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 499, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 306, + 512 + ], + "score": 1.0, + "content": "logits. This is surprising—the output logits are a", + "type": "text" + }, + { + "bbox": [ + 307, + 500, + 369, + 510 + ], + "score": 0.85, + "content": "1 1 3 \\cdot 1 1 3 \\cdot 1 1 3", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 499, + 505, + 512 + ], + "score": 1.0, + "content": "dimensional vector, but are well-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 104, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "approximated with just the 5 directions predicted by our interpretation. If we evaluate test loss using", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 520, + 488, + 533 + ], + "spans": [ + { + "bbox": [ + 104, + 520, + 389, + 533 + ], + "score": 1.0, + "content": "this logit approximation, we actually see an improvement in loss, from", + "type": "text" + }, + { + "bbox": [ + 389, + 521, + 431, + 532 + ], + "score": 0.89, + "content": "2 . 4 \\cdot 1 0 ^ { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 520, + 442, + 533 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 443, + 521, + 484, + 532 + ], + "score": 0.89, + "content": "4 . 7 \\cdot 1 0 ^ { - 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 520, + 488, + 533 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 466, + 506, + 533 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 538, + 504, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "Taken together, these results confirm that the model computes sums of terms of the form", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 548, + 426, + 563 + ], + "spans": [ + { + "bbox": [ + 107, + 549, + 423, + 561 + ], + "score": 0.84, + "content": "\\cos { \\left( w _ { k } ( \\bar { a } + b - c ) \\right) } = \\cos { \\left( w _ { k } ( a + b ) \\right) } \\cos { \\left( w _ { k } c \\right) } + \\sin { \\left( w _ { k } ( \\bar { a + } b ) \\right) } \\sin { \\left( w _ { k } c \\right) } .", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 548, + 426, + 563 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 106, + 538, + 505, + 563 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 573, + 425, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 573, + 426, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 426, + 586 + ], + "score": 1.0, + "content": "4.3 ZOOMING IN: APPROXIMATING NEURONS WITH SINES AND COSINES", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 593, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 452, + 606 + ], + "score": 1.0, + "content": "In the previous section, we showed how the model computes its final logits by using", + "type": "text" + }, + { + "bbox": [ + 452, + 594, + 468, + 605 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "to “read", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "off” trigonometric identities represented in the MLP neurons. We now examine the attention heads", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 617, + 504, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 504, + 627 + ], + "score": 1.0, + "content": "and MLP neurons to understand how the identities come to be represented at the MLP layer. In", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "Appendix C.1.2, we show that two of the attention heads approximately compute degree-2 poly-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "nomials of sines and cosines of a particular frequency (and the other two are used to increase the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "magnitude of the input embeddings in the residual stream). Here, we show that most neurons are", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "also well-approximated by degree-2 polynomials, and the map from neurons to logits is localized by", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 670, + 151, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 151, + 685 + ], + "score": 1.0, + "content": "frequency.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 594, + 505, + 685 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 688, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Most MLP neurons approximately compute a degree-2 polynomial of a single frequency. We", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "next try to approximate the activations of each MLP neuron by a degree-2 polynomial of one of the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 397, + 722 + ], + "score": 1.0, + "content": "5 key frequencies. As shown in Figure 5, out of 512 total neurons, 433", + "type": "text" + }, + { + "bbox": [ + 398, + 710, + 428, + 721 + ], + "score": 0.8, + "content": "( 8 4 . 6 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 709, + 473, + 722 + ], + "score": 1.0, + "content": ") have over", + "type": "text" + }, + { + "bbox": [ + 473, + 709, + 493, + 720 + ], + "score": 0.88, + "content": "8 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 302, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 302, + 734 + ], + "score": 1.0, + "content": "their variance explained with a single frequency.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 687, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 113, + 50, + 489, + 149 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 113, + 50, + 489, + 149 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 50, + 489, + 149 + ], + "spans": [ + { + "bbox": [ + 113, + 50, + 489, + 149 + ], + "score": 0.733, + "type": "image", + "image_path": "b954dc6f3ea4a22907512b5a4d0630472b633307a720388451f43dd306539259.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 113, + 50, + 489, + 83.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 113, + 83.0, + 489, + 116.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 113, + 116.0, + 489, + 149.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 157, + 505, + 190 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 156, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 505, + 171 + ], + "score": 1.0, + "content": "Figure 5: (Left) Most neurons are well-approximated by degree-2 polynomials of a single frequency.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 167, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 266, + 182 + ], + "score": 1.0, + "content": "(Right) A heatmap showing weights in", + "type": "text" + }, + { + "bbox": [ + 267, + 169, + 283, + 179 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 167, + 505, + 182 + ], + "score": 1.0, + "content": "corresponding to each of the 44 neurons of frequency", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 179, + 429, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 179, + 306, + 190 + ], + "score": 1.0, + "content": "14. The non-trivial components correspond to sin", + "type": "text" + }, + { + "bbox": [ + 306, + 179, + 327, + 189 + ], + "score": 0.39, + "content": "\\bar { ( \\iota _ { 1 1 } ) } _ { \\iota . 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 179, + 396, + 190 + ], + "score": 1.0, + "content": "and cos (wk) for", + "type": "text" + }, + { + "bbox": [ + 397, + 179, + 426, + 188 + ], + "score": 0.87, + "content": "k = 1 4", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 179, + 429, + 190 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "image", + "bbox": [ + 145, + 202, + 450, + 302 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 145, + 202, + 450, + 302 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 145, + 202, + 450, + 302 + ], + "spans": [ + { + "bbox": [ + 145, + 202, + 450, + 302 + ], + "score": 0.963, + "type": "image", + "image_path": "8dc1914cd85b773ec6c69f84dacf1232bec82e111d15eea61bcdf771f7a51812.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 145, + 202, + 450, + 235.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 145, + 235.33333333333334, + 450, + 268.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 145, + 268.6666666666667, + 450, + 302.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 312, + 505, + 356 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 311, + 504, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 433, + 325 + ], + "score": 1.0, + "content": "Figure 6: The loss of the transformer (lower=better) when ablating each frequency", + "type": "text" + }, + { + "bbox": [ + 434, + 311, + 504, + 324 + ], + "score": 0.92, + "content": "k \\in \\{ 1 , 2 , . . . , 5 6 \\}", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "and everything except for the five key frequencies (restricted loss). We include the original unablated", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 104, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "loss for reference. Ablating key frequencies causes a performance drop, while the other ablations", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 344, + 213, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 213, + 357 + ], + "score": 1.0, + "content": "do not harm performance.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + } + ], + "index": 8.75 + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "score": 1.0, + "content": "Maps to the logits are localized by frequency. We partition these 433 neurons by the frequencies", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 395, + 403 + ], + "score": 1.0, + "content": "with the highest variance explained. For each resulting subset, the map", + "type": "text" + }, + { + "bbox": [ + 395, + 391, + 412, + 402 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "from neurons to logits", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "has only two non-trivial components, corresponding to sine and cosine at that frequency. For exam-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 411, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 273, + 426 + ], + "score": 1.0, + "content": "ple, in Figure 5 we plot the 44 columns of", + "type": "text" + }, + { + "bbox": [ + 274, + 413, + 290, + 424 + ], + "score": 0.9, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 411, + 445, + 426 + ], + "score": 1.0, + "content": "corresponding to the 44 neurons in the", + "type": "text" + }, + { + "bbox": [ + 445, + 413, + 475, + 423 + ], + "score": 0.91, + "content": "k = 1 4", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 411, + 506, + 426 + ], + "score": 1.0, + "content": "cluster", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 103, + 419, + 470, + 440 + ], + "spans": [ + { + "bbox": [ + 103, + 419, + 319, + 440 + ], + "score": 1.0, + "content": "and find that the only non-negligible components are", + "type": "text" + }, + { + "bbox": [ + 320, + 423, + 359, + 437 + ], + "score": 0.83, + "content": "\\sin \\left( { \\frac { 2 \\overline { { k } } \\pi } { P } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 419, + 393, + 440 + ], + "score": 1.0, + "content": "and cos", + "type": "text" + }, + { + "bbox": [ + 394, + 423, + 418, + 437 + ], + "score": 0.9, + "content": "\\scriptstyle \\left( { \\frac { 2 k \\pi } { P } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 419, + 434, + 440 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 434, + 424, + 464, + 434 + ], + "score": 0.89, + "content": "k = 1 4", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 419, + 470, + 440 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 108, + 451, + 286, + 461 + ], + "lines": [ + { + "bbox": [ + 106, + 450, + 288, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 288, + 462 + ], + "score": 1.0, + "content": "4.4 CORRECTNESS CHECKS: ABLATIONS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 471, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 471, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 485 + ], + "score": 1.0, + "content": "In previous sections, we showed that various components of the model were well-approximated by", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 483, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 495 + ], + "score": 1.0, + "content": "sparse combinations of sines and cosines. We verify that these approximations are faithful to the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "model’s functionality, by replacing each component with its approximation. This generally does not", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 504, + 369, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 369, + 517 + ], + "score": 1.0, + "content": "hurt the performance of the model and in some cases improves it.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 521, + 504, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "MLP neurons. In Section 4.3, we identified 433 neurons that were well-approximated by a degree-2", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 544 + ], + "score": 1.0, + "content": "polynomial. We replace each of these neurons’ activation value by the corresponding polynomial,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 543, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 370, + 555 + ], + "score": 1.0, + "content": "leaving the other neurons untouched. This increases loss by only", + "type": "text" + }, + { + "bbox": [ + 370, + 543, + 385, + 554 + ], + "score": 0.86, + "content": "3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 543, + 479, + 555 + ], + "score": 1.0, + "content": "in relative terms (from", + "type": "text" + }, + { + "bbox": [ + 480, + 543, + 501, + 554 + ], + "score": 0.3, + "content": "2 . 4 1", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 543, + 505, + 555 + ], + "score": 1.0, + "content": "·", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 553, + 314, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 128, + 564 + ], + "score": 0.89, + "content": "1 0 ^ { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 553, + 140, + 567 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 140, + 554, + 187, + 565 + ], + "score": 0.9, + "content": "2 . 4 8 \\cdot 1 0 ^ { - 7 } \\phantom { }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 553, + 314, + 567 + ], + "score": 1.0, + "content": ") and has no effect on accuracy.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 570, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 106, + 571, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 505, + 583 + ], + "score": 1.0, + "content": "We can instead apply a stricter ablation to the MLP layer and restrict each neuron’s activation to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 582, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 420, + 594 + ], + "score": 1.0, + "content": "just the components of the polynomial corresponding to terms of the form", + "type": "text" + }, + { + "bbox": [ + 420, + 582, + 484, + 594 + ], + "score": 0.89, + "content": "\\cos ( w _ { k } ( a + b ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 582, + 506, + 594 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 168, + 605 + ], + "score": 0.92, + "content": "\\dot { \\sin ( \\boldsymbol { w } _ { k } ( a + \\boldsymbol { b } ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 592, + 355, + 605 + ], + "score": 1.0, + "content": "in the key frequencies. This improves loss by", + "type": "text" + }, + { + "bbox": [ + 356, + 593, + 375, + 604 + ], + "score": 0.86, + "content": "7 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 592, + 390, + 605 + ], + "score": 1.0, + "content": "(to", + "type": "text" + }, + { + "bbox": [ + 391, + 593, + 439, + 604 + ], + "score": 0.86, + "content": "5 . 5 4 \\cdot 1 0 ^ { - 8 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 592, + 506, + 605 + ], + "score": 1.0, + "content": ", validating that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 604, + 421, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 421, + 616 + ], + "score": 1.0, + "content": "the logits are calculated by trig identities of neurons as detailed in Section 4.2.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "Logit frequencies. Next, we ablate various components of the final logits in the Fourier space. To", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 631, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 237, + 645 + ], + "score": 1.0, + "content": "do so, we take a 2D DFT on the", + "type": "text" + }, + { + "bbox": [ + 237, + 632, + 298, + 642 + ], + "score": 0.81, + "content": "1 1 3 \\cdot 1 1 3 \\cdot 1 1 3", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 631, + 380, + 645 + ], + "score": 1.0, + "content": "logit matrix over all", + "type": "text" + }, + { + "bbox": [ + 380, + 632, + 419, + 642 + ], + "score": 0.64, + "content": "1 1 3 \\cdot 1 1 3", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 631, + 506, + 645 + ], + "score": 1.0, + "content": "pairs of inputs to get", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 643, + 410, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 410, + 655 + ], + "score": 1.0, + "content": "the logits in the Fourier basis, then set various frequencies in this basis to 0.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 660, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "We begin by ablating the components corresponding to each of the key frequencies. As reported in", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "score": 1.0, + "content": "Figure 6, ablating any key frequency causes a significant increase in loss. This confirms that the five", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "frequencies identified in previous sections are indeed necessary components of the transformer. In", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 693, + 370, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 370, + 704 + ], + "score": 1.0, + "content": "contrast, ablating other frequencies does not hurt the model at all.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 707, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 181, + 723 + ], + "score": 1.0, + "content": "We then ablate all", + "type": "text" + }, + { + "bbox": [ + 182, + 709, + 243, + 720 + ], + "score": 0.85, + "content": "1 1 3 \\cdot 1 1 3 - 4 0", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 707, + 505, + 723 + ], + "score": 1.0, + "content": "of the Fourier components besides key frequencies; this ablation", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 719, + 363, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 277, + 732 + ], + "score": 1.0, + "content": "actually improves performance (loss drops", + "type": "text" + }, + { + "bbox": [ + 278, + 720, + 298, + 731 + ], + "score": 0.88, + "content": "7 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 719, + 309, + 732 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 309, + 720, + 356, + 731 + ], + "score": 0.89, + "content": "7 . \\bar { 2 4 } \\cdot 1 0 ^ { - 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 719, + 363, + 732 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 113, + 50, + 489, + 149 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 113, + 50, + 489, + 149 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 50, + 489, + 149 + ], + "spans": [ + { + "bbox": [ + 113, + 50, + 489, + 149 + ], + "score": 0.733, + "type": "image", + "image_path": "b954dc6f3ea4a22907512b5a4d0630472b633307a720388451f43dd306539259.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 113, + 50, + 489, + 83.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 113, + 83.0, + 489, + 116.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 113, + 116.0, + 489, + 149.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 157, + 505, + 190 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 156, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 505, + 171 + ], + "score": 1.0, + "content": "Figure 5: (Left) Most neurons are well-approximated by degree-2 polynomials of a single frequency.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 167, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 266, + 182 + ], + "score": 1.0, + "content": "(Right) A heatmap showing weights in", + "type": "text" + }, + { + "bbox": [ + 267, + 169, + 283, + 179 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 167, + 505, + 182 + ], + "score": 1.0, + "content": "corresponding to each of the 44 neurons of frequency", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 179, + 429, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 179, + 306, + 190 + ], + "score": 1.0, + "content": "14. The non-trivial components correspond to sin", + "type": "text" + }, + { + "bbox": [ + 306, + 179, + 327, + 189 + ], + "score": 0.39, + "content": "\\bar { ( \\iota _ { 1 1 } ) } _ { \\iota . 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 179, + 396, + 190 + ], + "score": 1.0, + "content": "and cos (wk) for", + "type": "text" + }, + { + "bbox": [ + 397, + 179, + 426, + 188 + ], + "score": 0.87, + "content": "k = 1 4", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 179, + 429, + 190 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "image", + "bbox": [ + 145, + 202, + 450, + 302 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 145, + 202, + 450, + 302 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 145, + 202, + 450, + 302 + ], + "spans": [ + { + "bbox": [ + 145, + 202, + 450, + 302 + ], + "score": 0.963, + "type": "image", + "image_path": "8dc1914cd85b773ec6c69f84dacf1232bec82e111d15eea61bcdf771f7a51812.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 145, + 202, + 450, + 235.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 145, + 235.33333333333334, + 450, + 268.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 145, + 268.6666666666667, + 450, + 302.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 312, + 505, + 356 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 311, + 504, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 433, + 325 + ], + "score": 1.0, + "content": "Figure 6: The loss of the transformer (lower=better) when ablating each frequency", + "type": "text" + }, + { + "bbox": [ + 434, + 311, + 504, + 324 + ], + "score": 0.92, + "content": "k \\in \\{ 1 , 2 , . . . , 5 6 \\}", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "and everything except for the five key frequencies (restricted loss). We include the original unablated", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 104, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "loss for reference. Ablating key frequencies causes a performance drop, while the other ablations", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 344, + 213, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 213, + 357 + ], + "score": 1.0, + "content": "do not harm performance.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + } + ], + "index": 8.75 + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "score": 1.0, + "content": "Maps to the logits are localized by frequency. We partition these 433 neurons by the frequencies", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 395, + 403 + ], + "score": 1.0, + "content": "with the highest variance explained. For each resulting subset, the map", + "type": "text" + }, + { + "bbox": [ + 395, + 391, + 412, + 402 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "from neurons to logits", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "has only two non-trivial components, corresponding to sine and cosine at that frequency. For exam-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 411, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 273, + 426 + ], + "score": 1.0, + "content": "ple, in Figure 5 we plot the 44 columns of", + "type": "text" + }, + { + "bbox": [ + 274, + 413, + 290, + 424 + ], + "score": 0.9, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 411, + 445, + 426 + ], + "score": 1.0, + "content": "corresponding to the 44 neurons in the", + "type": "text" + }, + { + "bbox": [ + 445, + 413, + 475, + 423 + ], + "score": 0.91, + "content": "k = 1 4", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 411, + 506, + 426 + ], + "score": 1.0, + "content": "cluster", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 103, + 419, + 470, + 440 + ], + "spans": [ + { + "bbox": [ + 103, + 419, + 319, + 440 + ], + "score": 1.0, + "content": "and find that the only non-negligible components are", + "type": "text" + }, + { + "bbox": [ + 320, + 423, + 359, + 437 + ], + "score": 0.83, + "content": "\\sin \\left( { \\frac { 2 \\overline { { k } } \\pi } { P } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 419, + 393, + 440 + ], + "score": 1.0, + "content": "and cos", + "type": "text" + }, + { + "bbox": [ + 394, + 423, + 418, + 437 + ], + "score": 0.9, + "content": "\\scriptstyle \\left( { \\frac { 2 k \\pi } { P } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 419, + 434, + 440 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 434, + 424, + 464, + 434 + ], + "score": 0.89, + "content": "k = 1 4", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 419, + 470, + 440 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15, + "bbox_fs": [ + 103, + 378, + 506, + 440 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 451, + 286, + 461 + ], + "lines": [ + { + "bbox": [ + 106, + 450, + 288, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 288, + 462 + ], + "score": 1.0, + "content": "4.4 CORRECTNESS CHECKS: ABLATIONS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 471, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 471, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 485 + ], + "score": 1.0, + "content": "In previous sections, we showed that various components of the model were well-approximated by", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 483, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 495 + ], + "score": 1.0, + "content": "sparse combinations of sines and cosines. We verify that these approximations are faithful to the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "model’s functionality, by replacing each component with its approximation. This generally does not", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 504, + 369, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 369, + 517 + ], + "score": 1.0, + "content": "hurt the performance of the model and in some cases improves it.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 471, + 506, + 517 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 521, + 504, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "MLP neurons. In Section 4.3, we identified 433 neurons that were well-approximated by a degree-2", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 544 + ], + "score": 1.0, + "content": "polynomial. We replace each of these neurons’ activation value by the corresponding polynomial,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 543, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 370, + 555 + ], + "score": 1.0, + "content": "leaving the other neurons untouched. This increases loss by only", + "type": "text" + }, + { + "bbox": [ + 370, + 543, + 385, + 554 + ], + "score": 0.86, + "content": "3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 543, + 479, + 555 + ], + "score": 1.0, + "content": "in relative terms (from", + "type": "text" + }, + { + "bbox": [ + 480, + 543, + 501, + 554 + ], + "score": 0.3, + "content": "2 . 4 1", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 543, + 505, + 555 + ], + "score": 1.0, + "content": "·", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 553, + 314, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 128, + 564 + ], + "score": 0.89, + "content": "1 0 ^ { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 553, + 140, + 567 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 140, + 554, + 187, + 565 + ], + "score": 0.9, + "content": "2 . 4 8 \\cdot 1 0 ^ { - 7 } \\phantom { }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 553, + 314, + 567 + ], + "score": 1.0, + "content": ") and has no effect on accuracy.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 520, + 505, + 567 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 570, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 106, + 571, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 505, + 583 + ], + "score": 1.0, + "content": "We can instead apply a stricter ablation to the MLP layer and restrict each neuron’s activation to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 582, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 420, + 594 + ], + "score": 1.0, + "content": "just the components of the polynomial corresponding to terms of the form", + "type": "text" + }, + { + "bbox": [ + 420, + 582, + 484, + 594 + ], + "score": 0.89, + "content": "\\cos ( w _ { k } ( a + b ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 582, + 506, + 594 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 168, + 605 + ], + "score": 0.92, + "content": "\\dot { \\sin ( \\boldsymbol { w } _ { k } ( a + \\boldsymbol { b } ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 592, + 355, + 605 + ], + "score": 1.0, + "content": "in the key frequencies. This improves loss by", + "type": "text" + }, + { + "bbox": [ + 356, + 593, + 375, + 604 + ], + "score": 0.86, + "content": "7 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 592, + 390, + 605 + ], + "score": 1.0, + "content": "(to", + "type": "text" + }, + { + "bbox": [ + 391, + 593, + 439, + 604 + ], + "score": 0.86, + "content": "5 . 5 4 \\cdot 1 0 ^ { - 8 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 592, + 506, + 605 + ], + "score": 1.0, + "content": ", validating that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 604, + 421, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 421, + 616 + ], + "score": 1.0, + "content": "the logits are calculated by trig identities of neurons as detailed in Section 4.2.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 571, + 506, + 616 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "Logit frequencies. Next, we ablate various components of the final logits in the Fourier space. To", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 631, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 237, + 645 + ], + "score": 1.0, + "content": "do so, we take a 2D DFT on the", + "type": "text" + }, + { + "bbox": [ + 237, + 632, + 298, + 642 + ], + "score": 0.81, + "content": "1 1 3 \\cdot 1 1 3 \\cdot 1 1 3", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 631, + 380, + 645 + ], + "score": 1.0, + "content": "logit matrix over all", + "type": "text" + }, + { + "bbox": [ + 380, + 632, + 419, + 642 + ], + "score": 0.64, + "content": "1 1 3 \\cdot 1 1 3", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 631, + 506, + 645 + ], + "score": 1.0, + "content": "pairs of inputs to get", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 643, + 410, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 410, + 655 + ], + "score": 1.0, + "content": "the logits in the Fourier basis, then set various frequencies in this basis to 0.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 621, + 506, + 655 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 660, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "We begin by ablating the components corresponding to each of the key frequencies. As reported in", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "score": 1.0, + "content": "Figure 6, ablating any key frequency causes a significant increase in loss. This confirms that the five", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "frequencies identified in previous sections are indeed necessary components of the transformer. In", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 693, + 370, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 370, + 704 + ], + "score": 1.0, + "content": "contrast, ablating other frequencies does not hurt the model at all.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 659, + 505, + 704 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 707, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 181, + 723 + ], + "score": 1.0, + "content": "We then ablate all", + "type": "text" + }, + { + "bbox": [ + 182, + 709, + 243, + 720 + ], + "score": 0.85, + "content": "1 1 3 \\cdot 1 1 3 - 4 0", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 707, + 505, + 723 + ], + "score": 1.0, + "content": "of the Fourier components besides key frequencies; this ablation", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 719, + 363, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 277, + 732 + ], + "score": 1.0, + "content": "actually improves performance (loss drops", + "type": "text" + }, + { + "bbox": [ + 278, + 720, + 298, + 731 + ], + "score": 0.88, + "content": "7 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 719, + 309, + 732 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 309, + 720, + 356, + 731 + ], + "score": 0.89, + "content": "7 . \\bar { 2 4 } \\cdot 1 0 ^ { - 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 719, + 363, + 732 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 707, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 113, + 48, + 483, + 258 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 113, + 48, + 483, + 258 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 48, + 483, + 258 + ], + "spans": [ + { + "bbox": [ + 113, + 48, + 483, + 258 + ], + "score": 0.976, + "type": "image", + "image_path": "3ed053e601830d4d730c6d5e401e48b7c5f265a9bd65996c1eed6d55e594e6c2.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 113, + 48, + 483, + 118.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 113, + 118.0, + 483, + 188.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 113, + 188.0, + 483, + 258.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 268, + 505, + 357 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 267, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 282 + ], + "score": 1.0, + "content": "Figure 7: How each of the progress measures in Section 5.1 changes over the course of training.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 278, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 506, + 292 + ], + "score": 1.0, + "content": "The lines delineate the 3 phases of training: memorization, circuit formation, and cleanup (and a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 289, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 104, + 289, + 506, + 303 + ], + "score": 1.0, + "content": "final stable phase). (Top Left) Excluded loss increases during circuit formation, while train and test", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 302, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 314 + ], + "score": 1.0, + "content": "loss remain flat. (Top Right) The restricted loss begins declining before test loss declines, but has an", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 311, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 104, + 311, + 506, + 325 + ], + "score": 1.0, + "content": "inflection point when grokking begins to occur. (Bottom Left) The Gini coefficient of the norms of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 322, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 215, + 336 + ], + "score": 1.0, + "content": "the Fourier components of", + "type": "text" + }, + { + "bbox": [ + 215, + 324, + 232, + 334 + ], + "score": 0.9, + "content": "W _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 322, + 250, + 336 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 251, + 324, + 267, + 335 + ], + "score": 0.91, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 322, + 505, + 336 + ], + "score": 1.0, + "content": "increase sharply during cleanup. (Bottom Right) The sums", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 334, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 348 + ], + "score": 1.0, + "content": "of squared weights decreases smoothly during circuit formation and more sharply during cleanup,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 344, + 325, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 325, + 359 + ], + "score": 1.0, + "content": "indicating that both phases are linked to weight decay.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + } + ], + "index": 3.75 + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 165, + 392 + ], + "score": 1.0, + "content": "Directions in", + "type": "text" + }, + { + "bbox": [ + 165, + 380, + 181, + 391 + ], + "score": 0.88, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 380, + 313, + 392 + ], + "score": 1.0, + "content": ". In Section 4.2, we found that", + "type": "text" + }, + { + "bbox": [ + 313, + 380, + 330, + 391 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "is well approximated by the 10 directions", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 391, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 403 + ], + "score": 1.0, + "content": "corresponding to the cosine and sine of key frequencies. If we project the MLP activations to these", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 399, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 222, + 415 + ], + "score": 1.0, + "content": "10 directions, loss decreases", + "type": "text" + }, + { + "bbox": [ + 223, + 402, + 243, + 412 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 399, + 254, + 415 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 255, + 401, + 301, + 412 + ], + "score": 0.91, + "content": "1 . 1 9 \\cdot 1 0 ^ { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 399, + 506, + 415 + ], + "score": 1.0, + "content": ". If we instead projected the MLP activations onto", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "the nullspace of these 10 directions, loss increases to 5.27—worse than uniform. This suggests that", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 423, + 391, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 391, + 437 + ], + "score": 1.0, + "content": "the network achieves low loss using these and only these 10 directions.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 105, + 454, + 480, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 453, + 482, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 482, + 466 + ], + "score": 1.0, + "content": "5 UNDERSTANDING GROKKING BEHAVIOR USING PROGRESS MEASURES", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 479, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 492 + ], + "score": 1.0, + "content": "We now use our mechanistic understanding of the network to define two progress measures: metrics", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "score": 1.0, + "content": "that can be computed during training that track the progress of the model over the course of training,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "including during phase transitions. This allows us to study how the network reaches its final solution.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 108, + 528, + 226, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 227, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 227, + 540 + ], + "score": 1.0, + "content": "5.1 PROGRESS MEASURES", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 549, + 498, + 560 + ], + "lines": [ + { + "bbox": [ + 106, + 549, + 498, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 498, + 561 + ], + "score": 1.0, + "content": "We translate the ablations in Section 4.4 into two progress measures: restricted and excluded loss.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 566, + 504, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 393, + 579 + ], + "score": 1.0, + "content": "Restricted loss. Since the final network uses a sparse set of frequencies", + "type": "text" + }, + { + "bbox": [ + 393, + 568, + 406, + 577 + ], + "score": 0.86, + "content": "w _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 565, + 506, + 579 + ], + "score": 1.0, + "content": ", it makes sense to check", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "how well intermediate versions of the model can do using only those frequencies. To measure this,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 469, + 601 + ], + "score": 1.0, + "content": "we perform a 2D DFT on the logits to write them as a linear combination of waves in", + "type": "text" + }, + { + "bbox": [ + 469, + 590, + 475, + 598 + ], + "score": 0.69, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 588, + 496, + 601 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 496, + 588, + 501, + 598 + ], + "score": 0.68, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 588, + 505, + 601 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 422, + 612 + ], + "score": 1.0, + "content": "and set all terms besides the constant term and the 20 terms corresponding to", + "type": "text" + }, + { + "bbox": [ + 422, + 599, + 485, + 611 + ], + "score": 0.92, + "content": "\\cos ( w _ { k } ( a + b ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 610, + 500, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 168, + 622 + ], + "score": 0.92, + "content": "\\sin ( w _ { k } ( a + b ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 610, + 500, + 623 + ], + "score": 1.0, + "content": "for the five key frequencies to 0. We then measure the loss of the ablated network.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 504, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 357, + 640 + ], + "score": 1.0, + "content": "Excluded loss. Instead of keeping the important frequencies", + "type": "text" + }, + { + "bbox": [ + 358, + 629, + 371, + 638 + ], + "score": 0.87, + "content": "w _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 626, + 505, + 640 + ], + "score": 1.0, + "content": ", we next remove only those key", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "frequencies from the logits but keep the rest. We measure this on the training data to track how", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "much of the performance comes from Fourier multiplication versus memorization. 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If we instead projected the MLP activations onto", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "the nullspace of these 10 directions, loss increases to 5.27—worse than uniform. This suggests that", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 423, + 391, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 391, + 437 + ], + "score": 1.0, + "content": "the network achieves low loss using these and only these 10 directions.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 380, + 506, + 437 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 454, + 480, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 453, + 482, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 482, + 466 + ], + "score": 1.0, + "content": "5 UNDERSTANDING GROKKING BEHAVIOR USING PROGRESS MEASURES", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 479, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 492 + ], + "score": 1.0, + "content": "We now use our mechanistic understanding of the network to define two progress measures: metrics", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "score": 1.0, + "content": "that can be computed during training that track the progress of the model over the course of training,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "including during phase transitions. This allows us to study how the network reaches its final solution.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 478, + 506, + 514 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 528, + 226, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 227, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 227, + 540 + ], + "score": 1.0, + "content": "5.1 PROGRESS MEASURES", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 549, + 498, + 560 + ], + "lines": [ + { + "bbox": [ + 106, + 549, + 498, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 498, + 561 + ], + "score": 1.0, + "content": "We translate the ablations in Section 4.4 into two progress measures: restricted and excluded loss.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 106, + 549, + 498, + 561 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 566, + 504, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 393, + 579 + ], + "score": 1.0, + "content": "Restricted loss. 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We then measure the loss of the ablated network.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 565, + 506, + 623 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 504, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 357, + 640 + ], + "score": 1.0, + "content": "Excluded loss. Instead of keeping the important frequencies", + "type": "text" + }, + { + "bbox": [ + 358, + 629, + 371, + 638 + ], + "score": 0.87, + "content": "w _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 626, + 505, + 640 + ], + "score": 1.0, + "content": ", we next remove only those key", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "frequencies from the logits but keep the rest. We measure this on the training data to track how", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "much of the performance comes from Fourier multiplication versus memorization. The idea is that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 660, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 672 + ], + "score": 1.0, + "content": "the memorizing solution should be spread out in the Fourier domain, so that ablating a few directions", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 669, + 460, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 460, + 684 + ], + "score": 1.0, + "content": "will leave it mostly unaffected, while the generalizing solution will be hurt significantly.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 626, + 506, + 684 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Beyond these, we will also measure (1) the Gini coefficient (Hurley & Rickard, 2009) of the norms", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 231, + 711 + ], + "score": 1.0, + "content": "of the Fourier components of", + "type": "text" + }, + { + "bbox": [ + 231, + 699, + 248, + 710 + ], + "score": 0.92, + "content": "W _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 698, + 268, + 711 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 269, + 699, + 285, + 710 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 698, + 422, + 711 + ], + "score": 1.0, + "content": ", which measures the sparsity of", + "type": "text" + }, + { + "bbox": [ + 422, + 699, + 439, + 710 + ], + "score": 0.89, + "content": "W _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 698, + 460, + 711 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 460, + 699, + 477, + 710 + ], + "score": 0.91, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "in the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 211, + 723 + ], + "score": 1.0, + "content": "Fourier basis, and (2) the", + "type": "text" + }, + { + "bbox": [ + 211, + 710, + 221, + 721 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "-norm of the weights during training, since weight decay should push", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 721, + 279, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 279, + 732 + ], + "score": 1.0, + "content": "these down once the train loss is near zero.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 687, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 105, + 504, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "Using the mainline model from Section 4, we plot the excluded loss, restricted loss, Gini coefficient", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 115, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 169, + 129 + ], + "score": 1.0, + "content": "of the matrices", + "type": "text" + }, + { + "bbox": [ + 169, + 117, + 187, + 128 + ], + "score": 0.9, + "content": "W _ { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 115, + 206, + 129 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 206, + 117, + 222, + 127 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 115, + 505, + 129 + ], + "score": 1.0, + "content": ", and sum of squared weights in Figure 7. We find that training splits", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "into three phases, which we call the memorization, circuit formation, and cleanup phases. (We show", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 138, + 308, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 308, + 151 + ], + "score": 1.0, + "content": "similar results for other models in Appendix C.2.)", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 155, + 505, + 188 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "Memorization (Epochs 0k–1.4k). We first observe a decline of both excluded and train loss, with", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 166, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 505, + 178 + ], + "score": 1.0, + "content": "test and restricted loss both remaining high and the Gini coefficient staying relatively flat. In other", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 177, + 498, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 340, + 189 + ], + "score": 1.0, + "content": "words, the model memorizes the data, and the frequencies", + "type": "text" + }, + { + "bbox": [ + 340, + 178, + 353, + 188 + ], + "score": 0.86, + "content": "w _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 177, + 498, + 189 + ], + "score": 1.0, + "content": "used by the final model are unused.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 193, + 504, + 249 + ], + "lines": [ + { + "bbox": [ + 106, + 194, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 220, + 206 + ], + "score": 1.0, + "content": "Circuit formation (Epochs", + "type": "text" + }, + { + "bbox": [ + 221, + 194, + 262, + 204 + ], + "score": 0.39, + "content": "1 . 4 \\mathrm { k } { - 9 . 4 \\mathrm { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 194, + 505, + 206 + ], + "score": 1.0, + "content": "). In this phase, excluded loss rises, sum of squared weights", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "falls, restricted loss starts to fall, and test and train loss stay flat. This suggests that the model’s", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 215, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 506, + 228 + ], + "score": 1.0, + "content": "behavior on the train set transitions smoothly from the memorizing solution to the Fourier multi-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 227, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 240 + ], + "score": 1.0, + "content": "plication algorithm. The fall in the sum of squared weights suggests that circuit formation likely", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 237, + 462, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 462, + 251 + ], + "score": 1.0, + "content": "happens due to weight decay. Notably, the circuit is formed well before grokking occurs.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 254, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "score": 1.0, + "content": "Cleanup (Epochs 9.4k–14k). In this phase, excluded loss plateaus, restricted loss continues to drop,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 266, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 506, + 279 + ], + "score": 1.0, + "content": "test loss suddenly drops, and sum of squared weights sharply drops. As the completed Fourier", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "score": 1.0, + "content": "multiplication circuit both solves the task well and has lower weight than the memorization circuit,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 288, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 506, + 300 + ], + "score": 1.0, + "content": "weight decay encourages the network to shed the memorized solution in favor of focusing on the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "score": 1.0, + "content": "Fourier multiplication circuit. This is most cleanly shown in the sharp increase in the Gini coefficient", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 167, + 321 + ], + "score": 1.0, + "content": "for the matices", + "type": "text" + }, + { + "bbox": [ + 168, + 309, + 185, + 321 + ], + "score": 0.9, + "content": "W _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 310, + 203, + 321 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 203, + 309, + 219, + 321 + ], + "score": 0.9, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 310, + 505, + 321 + ], + "score": 1.0, + "content": ", which shows that the network is becoming sparser in the Fourier basis.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 108, + 339, + 271, + 351 + ], + "lines": [ + { + "bbox": [ + 105, + 338, + 273, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 273, + 353 + ], + "score": 1.0, + "content": "5.3 GROKKING AND WEIGHT DECAY", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "In the previous section, we saw that each phase of grokking corresponded to an inflection point in the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 373, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 107, + 374, + 116, + 384 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 373, + 505, + 386 + ], + "score": 1.0, + "content": "-norm of the weights. This suggests that weight decay is an important component of grokking and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "drives progress towards the generalizing solution. In Appendix D.1, we provide additional evidence", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "that weight decay is necessary for grokking: smaller amounts of weight decay causes the network", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "to take significantly longer to grok (echoing the results on toy models from Liu et al. (2022)), and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 418, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 429 + ], + "score": 1.0, + "content": "our networks do not grok on the modular arithmetic task without weight decay or some other form", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "of regularization. In Appendix C.2, we also find that the amount of data affects grokking: when", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "networks are provided with enough data, there is no longer a gap between the train and test losses", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 450, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 506, + 463 + ], + "score": 1.0, + "content": "(instead, both decline sharply some number of epochs into training). Finally, in Appendix D.3 we", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 461, + 353, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 353, + 474 + ], + "score": 1.0, + "content": "replicate these results on several additional algorithmic tasks.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 108, + 494, + 285, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 493, + 288, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 288, + 509 + ], + "score": 1.0, + "content": "6 CONCLUSION AND DISCUSSION", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 504, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "In this work, we use mechanistic interpretability to define progress measures for small transformers", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "trained on a modular addition task. We find that the transformers embed the input onto rotations", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 542, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 117, + 558 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 544, + 129, + 555 + ], + "score": 0.86, + "content": "\\mathbb { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 542, + 406, + 558 + ], + "score": 1.0, + "content": "and compose the rotations using trigonometric identities to compute", + "type": "text" + }, + { + "bbox": [ + 407, + 545, + 430, + 555 + ], + "score": 0.88, + "content": "a + b", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 542, + 506, + 558 + ], + "score": 1.0, + "content": "mod 113. Using", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "our reverse-engineered algorithm, we define two progress measures, along which the network makes", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "score": 1.0, + "content": "continuous progress toward the final algorithm prior to the grokking phase change. We see this work", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 577, + 477, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 477, + 590 + ], + "score": 1.0, + "content": "as a proof of concept for using mechanistic interpretability to understand emergent behavior.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 504, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "Larger models and realistic tasks. In this work, we studied the behavior of small transformers", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "on a simple algorithmic task, solved with a single circuit. On the other hand, larger models use", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "larger, more numerous circuits to solve significantly harder tasks (Cammarata et al., 2020; Wang", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "et al., 2022). The analysis reported in this work required significant amounts of manual effort, and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "our progress metrics are specific to small networks on one particular algorithmic task. Methods for", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 661 + ], + "score": 1.0, + "content": "automating the analysis and finding task-independent progress measures seem necessary to scale to", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 660, + 500, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 500, + 672 + ], + "score": 1.0, + "content": "other, larger models. We discuss possible scenarios for more realistic applications in Appendix F.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "Discovering phase change thresholds. While the progress measures we defined in Section 5.1 in-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "crease relatively smoothly before the phase transition (and suffice to allow us to understand grokking", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "for this task) we lack a general notion of criticality that would allow us to predict when the phase", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 708, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 724 + ], + "score": 1.0, + "content": "transition will happen ex ante. Future work should develop theory and practice in order to apply", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 354, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 354, + 733 + ], + "score": 1.0, + "content": "progress measures to predict the timing of emergent behavior.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 112, + 83, + 459, + 93 + ], + "lines": [ + { + "bbox": [ + 111, + 82, + 461, + 95 + ], + "spans": [ + { + "bbox": [ + 111, + 82, + 461, + 95 + ], + "score": 1.0, + "content": ".2 PHASES OF GROKKING: MEMORIZATION, CIRCUIT FORMATION, AND CLEANUP", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 105, + 504, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "Using the mainline model from Section 4, we plot the excluded loss, restricted loss, Gini coefficient", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 115, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 169, + 129 + ], + "score": 1.0, + "content": "of the matrices", + "type": "text" + }, + { + "bbox": [ + 169, + 117, + 187, + 128 + ], + "score": 0.9, + "content": "W _ { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 115, + 206, + 129 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 206, + 117, + 222, + 127 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 115, + 505, + 129 + ], + "score": 1.0, + "content": ", and sum of squared weights in Figure 7. We find that training splits", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "into three phases, which we call the memorization, circuit formation, and cleanup phases. (We show", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 138, + 308, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 308, + 151 + ], + "score": 1.0, + "content": "similar results for other models in Appendix C.2.)", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 105, + 505, + 151 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 155, + 505, + 188 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "Memorization (Epochs 0k–1.4k). We first observe a decline of both excluded and train loss, with", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 166, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 505, + 178 + ], + "score": 1.0, + "content": "test and restricted loss both remaining high and the Gini coefficient staying relatively flat. In other", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 177, + 498, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 340, + 189 + ], + "score": 1.0, + "content": "words, the model memorizes the data, and the frequencies", + "type": "text" + }, + { + "bbox": [ + 340, + 178, + 353, + 188 + ], + "score": 0.86, + "content": "w _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 177, + 498, + 189 + ], + "score": 1.0, + "content": "used by the final model are unused.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 154, + 505, + 189 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 193, + 504, + 249 + ], + "lines": [ + { + "bbox": [ + 106, + 194, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 220, + 206 + ], + "score": 1.0, + "content": "Circuit formation (Epochs", + "type": "text" + }, + { + "bbox": [ + 221, + 194, + 262, + 204 + ], + "score": 0.39, + "content": "1 . 4 \\mathrm { k } { - 9 . 4 \\mathrm { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 194, + 505, + 206 + ], + "score": 1.0, + "content": "). In this phase, excluded loss rises, sum of squared weights", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "falls, restricted loss starts to fall, and test and train loss stay flat. This suggests that the model’s", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 215, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 506, + 228 + ], + "score": 1.0, + "content": "behavior on the train set transitions smoothly from the memorizing solution to the Fourier multi-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 227, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 240 + ], + "score": 1.0, + "content": "plication algorithm. The fall in the sum of squared weights suggests that circuit formation likely", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 237, + 462, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 462, + 251 + ], + "score": 1.0, + "content": "happens due to weight decay. Notably, the circuit is formed well before grokking occurs.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 194, + 506, + 251 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 254, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "score": 1.0, + "content": "Cleanup (Epochs 9.4k–14k). In this phase, excluded loss plateaus, restricted loss continues to drop,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 266, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 506, + 279 + ], + "score": 1.0, + "content": "test loss suddenly drops, and sum of squared weights sharply drops. As the completed Fourier", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "score": 1.0, + "content": "multiplication circuit both solves the task well and has lower weight than the memorization circuit,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 288, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 506, + 300 + ], + "score": 1.0, + "content": "weight decay encourages the network to shed the memorized solution in favor of focusing on the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "score": 1.0, + "content": "Fourier multiplication circuit. This is most cleanly shown in the sharp increase in the Gini coefficient", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 167, + 321 + ], + "score": 1.0, + "content": "for the matices", + "type": "text" + }, + { + "bbox": [ + 168, + 309, + 185, + 321 + ], + "score": 0.9, + "content": "W _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 310, + 203, + 321 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 203, + 309, + 219, + 321 + ], + "score": 0.9, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 310, + 505, + 321 + ], + "score": 1.0, + "content": ", which shows that the network is becoming sparser in the Fourier basis.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 254, + 506, + 321 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 339, + 271, + 351 + ], + "lines": [ + { + "bbox": [ + 105, + 338, + 273, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 273, + 353 + ], + "score": 1.0, + "content": "5.3 GROKKING AND WEIGHT DECAY", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "In the previous section, we saw that each phase of grokking corresponded to an inflection point in the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 373, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 107, + 374, + 116, + 384 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 373, + 505, + 386 + ], + "score": 1.0, + "content": "-norm of the weights. This suggests that weight decay is an important component of grokking and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "drives progress towards the generalizing solution. In Appendix D.1, we provide additional evidence", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "that weight decay is necessary for grokking: smaller amounts of weight decay causes the network", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "to take significantly longer to grok (echoing the results on toy models from Liu et al. (2022)), and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 418, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 429 + ], + "score": 1.0, + "content": "our networks do not grok on the modular arithmetic task without weight decay or some other form", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "of regularization. In Appendix C.2, we also find that the amount of data affects grokking: when", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "networks are provided with enough data, there is no longer a gap between the train and test losses", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 450, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 506, + 463 + ], + "score": 1.0, + "content": "(instead, both decline sharply some number of epochs into training). Finally, in Appendix D.3 we", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 461, + 353, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 353, + 474 + ], + "score": 1.0, + "content": "replicate these results on several additional algorithmic tasks.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 362, + 506, + 474 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 494, + 285, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 493, + 288, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 288, + 509 + ], + "score": 1.0, + "content": "6 CONCLUSION AND DISCUSSION", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 504, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "In this work, we use mechanistic interpretability to define progress measures for small transformers", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "trained on a modular addition task. We find that the transformers embed the input onto rotations", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 542, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 117, + 558 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 544, + 129, + 555 + ], + "score": 0.86, + "content": "\\mathbb { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 542, + 406, + 558 + ], + "score": 1.0, + "content": "and compose the rotations using trigonometric identities to compute", + "type": "text" + }, + { + "bbox": [ + 407, + 545, + 430, + 555 + ], + "score": 0.88, + "content": "a + b", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 542, + 506, + 558 + ], + "score": 1.0, + "content": "mod 113. Using", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "our reverse-engineered algorithm, we define two progress measures, along which the network makes", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "score": 1.0, + "content": "continuous progress toward the final algorithm prior to the grokking phase change. We see this work", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 577, + 477, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 477, + 590 + ], + "score": 1.0, + "content": "as a proof of concept for using mechanistic interpretability to understand emergent behavior.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 522, + 506, + 590 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 504, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "Larger models and realistic tasks. In this work, we studied the behavior of small transformers", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "on a simple algorithmic task, solved with a single circuit. On the other hand, larger models use", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "larger, more numerous circuits to solve significantly harder tasks (Cammarata et al., 2020; Wang", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "et al., 2022). The analysis reported in this work required significant amounts of manual effort, and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "our progress metrics are specific to small networks on one particular algorithmic task. Methods for", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 661 + ], + "score": 1.0, + "content": "automating the analysis and finding task-independent progress measures seem necessary to scale to", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 660, + 500, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 500, + 672 + ], + "score": 1.0, + "content": "other, larger models. We discuss possible scenarios for more realistic applications in Appendix F.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 594, + 506, + 672 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "Discovering phase change thresholds. 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The", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 116, + 275, + 504, + 289 + ], + "spans": [ + { + "bbox": [ + 116, + 275, + 504, + 289 + ], + "score": 1.0, + "content": "slingshot mechanism: An empirical study of adaptive optimizers and the grokking phenomenon.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 115, + 286, + 279, + 298 + ], + "spans": [ + { + "bbox": [ + 115, + 286, + 279, + 298 + ], + "score": 1.0, + "content": "arXiv preprint arXiv:2206.04817, 2022.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 106, + 264, + 505, + 298 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 305, + 505, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 306, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 505, + 318 + ], + "score": 1.0, + "content": "Kevin Wang, Alexandre Variengien, Arthur Conmy, Buck Shlegeris, and Jacob Steinhardt. Inter-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 115, + 316, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 115, + 316, + 506, + 329 + ], + "score": 1.0, + "content": "pretability in the wild: a circuit for indirect object identification in gpt-2 small. arXiv preprint", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 115, + 327, + 219, + 339 + ], + "spans": [ + { + "bbox": [ + 115, + 327, + 219, + 339 + ], + "score": 1.0, + "content": "arXiv:2211.00593, 2022.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 106, + 306, + 506, + 339 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 345, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 346, + 504, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 504, + 358 + ], + "score": 1.0, + "content": "Jason Wei, Yi Tay, Rishi Bommasani, Colin Raffel, Barret Zoph, Sebastian Borgeaud, Dani Yo-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 116, + 357, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 116, + 357, + 505, + 371 + ], + "score": 1.0, + "content": "gatama, Maarten Bosma, Denny Zhou, Donald Metzler, et al. Emergent abilities of large language", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 116, + 369, + 317, + 381 + ], + "spans": [ + { + "bbox": [ + 116, + 369, + 317, + 381 + ], + "score": 1.0, + "content": "models. arXiv preprint arXiv:2206.07682, 2022a.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 346, + 505, + 381 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 387, + 504, + 421 + ], + "lines": [ + { + "bbox": [ + 104, + 385, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 104, + 385, + 505, + 401 + ], + "score": 1.0, + "content": "Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Ed Chi, Quoc Le, and Denny", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 115, + 397, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 115, + 397, + 506, + 411 + ], + "score": 1.0, + "content": "Zhou. 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In Stefan van der Walt and ´", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 116, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 116, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "Jarrod Millman (eds.), Proceedings of the 9th Python in Science Conference, pp. 56 – 61, 2010.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 116, + 450, + 269, + 462 + ], + "spans": [ + { + "bbox": [ + 116, + 450, + 269, + 462 + ], + "score": 1.0, + "content": "doi: 10.25080/Majora-92bf1922-00a.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 427, + 506, + 462 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 400, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 401, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 401, + 96 + ], + "score": 1.0, + "content": "A MATHEMATICAL STRUCTURE OF THE TRANSFORMER", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 108, + 106, + 503, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 104, + 505, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 120 + ], + "score": 1.0, + "content": "We follow the conventions and notation of Elhage et al. (2021) in describing our model. Here, we", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 117, + 357, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 357, + 129 + ], + "score": 1.0, + "content": "briefly recap their notation and examine it in our specific case.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 133, + 505, + 178 + ], + "lines": [ + { + "bbox": [ + 105, + 132, + 506, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 283, + 147 + ], + "score": 1.0, + "content": "We denote our hyperparameters as follows:", + "type": "text" + }, + { + "bbox": [ + 284, + 134, + 339, + 145 + ], + "score": 0.92, + "content": "d _ { v o c a b } = 1 1 3", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 132, + 506, + 147 + ], + "score": 1.0, + "content": "is the size of the input and output spaces", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 145, + 504, + 156 + ], + "spans": [ + { + "bbox": [ + 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Note that we do not have", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "biases in our embedding, attention layer or unembedding, and we do not tie the matrices for the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 241, + 239, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 239, + 255 + ], + "score": 1.0, + "content": "embedding/unembedding layers.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 259, + 505, + 353 + ], + "lines": [ + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "We now describe the mathematical structure of our network. Note that loss is only calculated from", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 270, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 283 + ], + "score": 1.0, + "content": "the logits on the final token, and information only moves between tokens during the attention layer,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "score": 1.0, + "content": "so our variables from the end of the attention layer onwards only refer to the final token. We use", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 292, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 114, + 304 + ], + "score": 0.84, + "content": "t _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 292, + 245, + 305 + ], + "score": 1.0, + "content": "to denote the token in position", + "type": "text" + }, + { + "bbox": [ + 246, + 293, + 251, + 302 + ], + "score": 0.62, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 292, + 380, + 305 + ], + "score": 1.0, + "content": "(as a one-hot encoded vector),", + "type": "text" + }, + { + "bbox": [ + 380, + 294, + 390, + 304 + ], + "score": 0.83, + "content": "p _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 292, + 505, + 305 + ], + "score": 1.0, + "content": "to denote the ith positional", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 303, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 171, + 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the MLP layer on the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 340, + 486, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 154, + 354 + ], + "score": 1.0, + "content": "final token,", + "type": "text" + }, + { + "bbox": [ + 154, + 340, + 171, + 351 + ], + "score": 0.89, + "content": "x ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 340, + 486, + 354 + ], + "score": 1.0, + "content": "the final residual stream on the final token, Logits the logits on the final token.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 358, + 319, + 370 + ], + "lines": [ + { + "bbox": [ + 105, + 356, + 321, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 321, + 372 + ], + "score": 1.0, + "content": "The logits are calculated via the following equations:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 389, + 420, + 505 + ], + "lines": [ + { + "bbox": [ + 194, + 389, + 420, + 505 + ], + "spans": [ + { + "bbox": [ + 194, + 389, + 420, + 505 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { x _ { i } ^ { ( 0 ) } = W _ { E } t _ { i } + p _ { i } } \\\\ & { \\begin{array} { r l } & { A ^ { j } = \\mathrm { s o f t m a x } ( x ^ { ( 0 ) ^ { T } } W _ { K } ^ { j ^ { T } } W _ { Q } ^ { j } x _ { 2 } ^ { ( 0 ) } ) } \\\\ & { x ^ { ( 1 ) } = [ \\displaystyle \\sum _ { j } W _ { Q } ^ { j } W _ { V } ^ { j } ( x ^ { ( 0 ) } \\cdot A ^ { j } ) ] + x _ { 2 } ^ { ( 0 ) } } \\end{array} } \\\\ & { \\begin{array} { r l } & { \\mathrm { M L P } = \\mathrm { R e L U } ( W _ { i n } x ^ { ( 1 ) } ) } \\\\ & { x ^ { ( 2 ) } = W _ { o u t } N + x ^ { ( 1 ) } = W _ { o u t } \\mathrm { R e L U } ( W _ { i n } x ^ { ( 1 ) } ) + x ^ { ( 1 ) } } \\end{array} } \\\\ & { \\begin{array} { r l } & { \\mathrm { c o p i t s } = W _ { U L } x ^ { ( 2 ) } } \\end{array} } \\end{array}", + 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"spans": [ + { + "bbox": [ + 104, + 514, + 295, + 532 + ], + "score": 1.0, + "content": "As in Elhage et al. (2021), we refer to the term", + "type": "text" + }, + { + "bbox": [ + 295, + 516, + 353, + 531 + ], + "score": 0.93, + "content": "W _ { O } ^ { j } W _ { V } ^ { j } ( x ^ { ( 0 ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 514, + 459, + 532 + ], + "score": 1.0, + "content": "as the OV circuit for head", + "type": "text" + }, + { + "bbox": [ + 459, + 519, + 465, + 529 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 108, + 542, + 295, + 554 + ], + "lines": [ + { + "bbox": [ + 106, + 542, + 296, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 296, + 555 + ], + "score": 1.0, + "content": "A.1 EMPIRICAL MODEL SIMPLIFICATIONS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 108, + 563, + 257, + 575 + ], + "lines": [ + { + "bbox": [ + 106, + 562, + 259, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 259, + 577 + ], + "score": 1.0, + "content": "We make two empirical observations:", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 133, + 580, + 505, + 701 + ], + "lines": [ + { + "bbox": [ + 132, + 579, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 132, + 579, + 242, + 593 + ], + "score": 1.0, + "content": "• The attention paid from", + "type": "text" + }, + { + "bbox": [ + 243, + 581, + 259, + 591 + ], + "score": 0.8, + "content": "\" = \"", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 579, + 506, + 593 + ], + "score": 1.0, + "content": "to itself is trivial. In practice, the average attention paid is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 142, + 591, + 492, + 604 + ], + "spans": [ + { + "bbox": [ + 142, + 592, + 164, + 602 + ], + "score": 0.85, + "content": "0 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 591, + 175, + 604 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 176, + 592, + 198, + 602 + ], + "score": 0.86, + "content": "0 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 591, + 492, + 604 + ], + "score": 1.0, + "content": "for each head, and ablating this does not affect model performance at all.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 134, + 601, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 134, + 601, + 506, + 614 + ], + "score": 1.0, + "content": "• The skip connection around the MLP layer is not important for the model’s computation", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 613, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 141, + 613, + 506, + 626 + ], + "score": 1.0, + "content": "and can be ignored. 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This is a significant increase in loss, but from such a small baseline that", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 141, + 646, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 141, + 646, + 506, + 658 + ], + "score": 1.0, + "content": "we can still ignore it and reverse engineer the model’s computation. (That being said, both", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 141, + 658, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 141, + 658, + 506, + 669 + ], + "score": 1.0, + "content": "the attention heads and the skip connection around them are crucial to the functioning of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 141, + 667, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 141, + 667, + 506, + 681 + ], + "score": 1.0, + "content": "the model: zero ablating attention heads increases loss to 24.3, while zero ablating the skip", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 141, + 679, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 141, + 679, + 505, + 691 + ], + "score": 1.0, + "content": "connection around the attention heads increases loss to 19.1, both significantly worse than", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 141, + 689, + 179, + 703 + ], + "spans": [ + { + "bbox": [ + 141, + 689, + 179, + 703 + ], + "score": 1.0, + "content": "chance.)", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 108, + 707, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 107, + 707, + 505, + 719 + ], + "spans": [ + { + "bbox": [ + 107, + 707, + 505, + 719 + ], + "score": 1.0, + "content": "A consequence of the first observation is that the attention is now a softmax over 2 elements, i.e. a", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 103, + 717, + 505, + 736 + ], + "spans": [ + { + "bbox": [ + 103, + 717, + 240, + 736 + ], + "score": 1.0, + "content": "sigmoid over the difference. 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Note that we do not have", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "biases in our embedding, attention layer or unembedding, and we do not tie the matrices for the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 241, + 239, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 239, + 255 + ], + "score": 1.0, + "content": "embedding/unembedding layers.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 181, + 507, + 255 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 259, + 505, + 353 + ], + "lines": [ + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "We now describe the mathematical structure of our network. Note that loss is only calculated from", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 270, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 283 + ], + "score": 1.0, + "content": "the logits on the final token, and information only moves between tokens during the attention layer,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "score": 1.0, + "content": "so our variables from the end of the attention layer onwards only refer to the final token. We use", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 292, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 114, + 304 + ], + "score": 0.84, + "content": "t _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 292, + 245, + 305 + ], + "score": 1.0, + "content": "to denote the token in position", + "type": "text" + }, + { + "bbox": [ + 246, + 293, + 251, + 302 + ], + "score": 0.62, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 292, + 380, + 305 + ], + "score": 1.0, + "content": "(as a one-hot encoded vector),", + "type": "text" + }, + { + "bbox": [ + 380, + 294, + 390, + 304 + ], + "score": 0.83, + "content": "p _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 292, + 505, + 305 + ], + "score": 1.0, + "content": "to denote the ith positional", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 303, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 171, + 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the MLP layer on the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 340, + 486, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 154, + 354 + ], + "score": 1.0, + "content": "final token,", + "type": "text" + }, + { + "bbox": [ + 154, + 340, + 171, + 351 + ], + "score": 0.89, + "content": "x ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 340, + 486, + 354 + ], + "score": 1.0, + "content": "the final residual stream on the final token, Logits the logits on the final token.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 259, + 505, + 354 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 358, + 319, + 370 + ], + "lines": [ + { + "bbox": [ + 105, + 356, + 321, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 321, + 372 + ], + "score": 1.0, + "content": "The logits are calculated via the following equations:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 356, + 321, + 372 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 389, + 420, + 505 + ], + "lines": [ + { + "bbox": [ + 194, + 389, + 420, + 505 + ], + "spans": [ + { + "bbox": [ + 194, + 389, + 420, + 505 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { x _ { i } ^ { ( 0 ) } = W _ { E } t _ { i } + p _ { i } } \\\\ & { \\begin{array} { r l } & { A ^ { j } = \\mathrm { s o f t m a x } ( x ^ { ( 0 ) ^ { T } } W _ { K } ^ { j ^ { T } } W _ { Q } ^ { j } x _ { 2 } ^ { ( 0 ) } ) } \\\\ & { x ^ { ( 1 ) } = [ \\displaystyle \\sum _ { j } W _ { Q } ^ { j } W _ { V } ^ { j } ( x ^ { ( 0 ) } \\cdot A ^ { j } ) ] + x _ { 2 } ^ { ( 0 ) } } \\end{array} } \\\\ & { \\begin{array} { r l } & { \\mathrm { M L P } = \\mathrm { R e L U } ( W _ { i n } x ^ { ( 1 ) } ) } \\\\ & { x ^ { ( 2 ) } = W _ { o u t } N + x ^ { ( 1 ) } = W _ { o u t } \\mathrm { R e L U } ( W _ { i n } x ^ { ( 1 ) } ) + x ^ { ( 1 ) } } \\end{array} } \\\\ & { \\begin{array} { r l } & { \\mathrm { c o p i t s } = W _ { U L } x ^ { ( 2 ) } } \\end{array} } \\end{array}", + "type": "interline_equation", + "image_path": "cfef52e41dbc44e831a5477e54f8271001f36902318b1fb2997845a3f1a07fa1.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 194, + 389, + 420, + 403.5 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 194, + 403.5, + 420, + 418.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 194, + 418.0, + 420, + 432.5 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 194, + 432.5, + 420, + 447.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 194, + 447.0, + 420, + 461.5 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 194, + 461.5, + 420, + 476.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 194, + 476.0, + 420, + 490.5 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 194, + 490.5, + 420, + 505.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 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(2021), we refer to the term", + "type": "text" + }, + { + "bbox": [ + 295, + 516, + 353, + 531 + ], + "score": 0.93, + "content": "W _ { O } ^ { j } W _ { V } ^ { j } ( x ^ { ( 0 ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 514, + 459, + 532 + ], + "score": 1.0, + "content": "as the OV circuit for head", + "type": "text" + }, + { + "bbox": [ + 459, + 519, + 465, + 529 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 104, + 514, + 465, + 532 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 542, + 295, + 554 + ], + "lines": [ + { + "bbox": [ + 106, + 542, + 296, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 296, + 555 + ], + "score": 1.0, + "content": "A.1 EMPIRICAL MODEL SIMPLIFICATIONS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 106, + 542, + 296, + 555 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 563, + 257, + 575 + ], + "lines": [ + { + "bbox": [ + 106, + 562, + 259, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 259, + 577 + ], + "score": 1.0, + "content": "We make two empirical observations:", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 106, + 562, + 259, + 577 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 580, + 505, + 701 + ], + "lines": [ + { + "bbox": [ + 132, + 579, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 132, + 579, + 242, + 593 + ], + "score": 1.0, + "content": "• The attention paid from", + "type": "text" + }, + { + "bbox": [ + 243, + 581, + 259, + 591 + ], + "score": 0.8, + "content": "\" = \"", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 579, + 506, + 593 + ], + "score": 1.0, + "content": "to itself is trivial. In practice, the average attention paid is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 142, + 591, + 492, + 604 + ], + "spans": [ + { + "bbox": [ + 142, + 592, + 164, + 602 + ], + "score": 0.85, + "content": "0 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 591, + 175, + 604 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 176, + 592, + 198, + 602 + ], + "score": 0.86, + "content": "0 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 591, + 492, + 604 + ], + "score": 1.0, + "content": "for each head, and ablating this does not affect model performance at all.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 134, + 601, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 134, + 601, + 506, + 614 + ], + "score": 1.0, + "content": "• The skip connection around the MLP layer is not important for the model’s computation", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 613, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 141, + 613, + 506, + 626 + ], + "score": 1.0, + "content": "and can be ignored. 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So", + "type": "text" + }, + { + "bbox": [ + 295, + 332, + 458, + 358 + ], + "score": 0.94, + "content": "A _ { 0 } ^ { j } = \\sigma \\left( { x _ { 2 } ^ { ( 0 ) } } ^ { T } { W _ { Q } ^ { j } } ^ { T } W _ { K } ( x _ { 0 } ^ { ( 0 ) } - x _ { 1 } ^ { ( 0 ) } ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 328, + 479, + 359 + ], + "score": 1.0, + "content": "(and", + "type": "text" + }, + { + "bbox": [ + 480, + 336, + 506, + 352 + ], + "score": 0.82, + "content": "A _ { 1 } ^ { j } =", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 356, + 141, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 141, + 372 + ], + "score": 0.8, + "content": "1 - A _ { 0 } ^ { j } )", + "type": "inline_equation", + "image_path": "54654314f6bfa6be7f148afe966c9226b2cad95ff94eb3b88fbb4983fca6d00d.jpg" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 375, + 506, + 409 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 342, + 389 + ], + "score": 1.0, + "content": "A consequence of the second observation is that Logits", + "type": "text" + }, + { + "bbox": [ + 342, + 376, + 415, + 387 + ], + "score": 0.9, + "content": "\\approx { W _ { U } } { W _ { o u t } } { \\mathrm { M L P } }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 374, + 506, + 389 + ], + "score": 1.0, + "content": ", which we denote as", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 387, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 107, + 387, + 175, + 398 + ], + "score": 0.92, + "content": "W _ { L } = W _ { U } W _ { o u t }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 387, + 331, + 399 + ], + "score": 1.0, + "content": ". 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The reason for this is to end up with a function whose", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 486, + 367, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 140, + 498 + ], + "score": 1.0, + "content": "value at", + "type": "text" + }, + { + "bbox": [ + 140, + 487, + 166, + 497 + ], + "score": 0.86, + "content": "x = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 486, + 356, + 498 + ], + "score": 1.0, + "content": "mod 113 is significantly larger than any other", + "type": "text" + }, + { + "bbox": [ + 356, + 489, + 363, + 496 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 486, + 367, + 498 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 106, + 464, + 505, + 498 + ] + }, + { + "type": "list", + "bbox": [ + 107, + 502, + 505, + 537 + ], + "lines": [ + { + "bbox": [ + 105, + 495, + 510, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 164, + 524 + ], + "score": 1.0, + "content": "For example,", + "type": "text" + }, + { + "bbox": [ + 193, + 495, + 253, + 524 + ], + "score": 1.0, + "content": "er the function", + "type": "text" + }, + { + "bbox": [ + 253, + 502, + 348, + 516 + ], + "score": 0.93, + "content": "\\begin{array} { r } { f _ { 1 4 } ( x ) = \\cos \\left( \\frac { 2 \\pi \\cdot 1 4 } { 1 1 3 } x \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 495, + 510, + 524 + ], + "score": 1.0, + "content": "This function has period 113 and is", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 164, + 514, + 355, + 525 + ], + "spans": [ + { + "bbox": [ + 164, + 515, + 193, + 524 + ], + "score": 0.87, + "content": "x = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 514, + 348, + 525 + ], + "score": 1.0, + "content": "mod 113. However, other values of", + "type": "text" + }, + { + "bbox": [ + 348, + 516, + 355, + 524 + ], + "score": 0.78, + "content": "x", + "type": "inline_equation" + } + ], + "index": 18, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 525, + 356, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 219, + 538 + ], + "score": 0.88, + "content": "f _ { 1 4 } ( 8 ) = f _ { 1 4 } ( 1 0 5 ) = 0 . 9 9 8", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 525, + 223, + 538 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 223, + 525, + 336, + 538 + ], + "score": 0.86, + "content": "f _ { 1 4 } ( 1 6 ) = f _ { 1 4 } ( 8 9 ) = 0 . 9 9 4", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 525, + 356, + 538 + ], + "score": 1.0, + "content": ", etc.", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 18, + "bbox_fs": [ + 105, + 495, + 510, + 538 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 541, + 505, + 600 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 507, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 165, + 558 + ], + "score": 1.0, + "content": "Now consider", + "type": "text" + }, + { + "bbox": [ + 166, + 542, + 260, + 556 + ], + "score": 0.92, + "content": "\\begin{array} { r } { f _ { 3 5 } ( x ) = \\cos { \\left( \\frac { 2 \\pi \\cdot 3 5 } { 1 1 3 } x \\right) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 538, + 507, + 558 + ], + "score": 1.0, + "content": ". 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1.0, + "content": "token at the first sequence position, as a heatmap over the inputs. All four attention heads exhibit", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 206, + 187, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 187, + 219 + ], + "score": 1.0, + "content": "striking periodicity.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 231, + 388, + 286 + ], + "lines": [ + { + "bbox": [ + 221, + 231, + 386, + 286 + ], + "spans": [ + { + "bbox": [ + 221, + 231, + 386, + 286 + ], + "score": 0.43, + "content": "\\begin{array}{c} \\begin{array} { r l r } & { } & { \\frac { \\mathrm { H e a d } } { 0 } \\parallel \\boldsymbol { k } \\parallel \\quad \\begin{array} { c } { \\alpha ^ { j } } \\\\ { 3 5 } \\end{array} | \\quad \\frac { \\alpha ^ { j } } { - 0 . 2 6 } \\parallel \\quad \\begin{array} { c } { \\beta ^ { j } } \\\\ { - 0 . 1 4 } \\end{array} | \\quad \\mathrm { H V E } } \\\\ & { } & { 1 } \\\\ & { } & { 2 } \\\\ & { } & { 3 } \\end{array} | \\begin{array} { l } { 4 2 } \\\\ { 5 2 } \\\\ { 4 2 } \\end{array} | \\quad 0 . 2 7 \\parallel \\quad \\begin{array} { c } { \\beta ^ { j } } \\\\ { 0 . 0 4 } \\end{array} | \\quad 9 9 . 0 3 \\% \\end{array}", + "type": "interline_equation", + "image_path": "265a5ccc87ded124f4a0d94f8dd39157688a00103723e08ef07159bb1c430b34.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 221, + 231, + 388, + 244.75 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 221, + 244.75, + 388, + 258.5 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 221, + 258.5, + 388, + 272.25 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 221, + 272.25, + 388, + 286.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 293, + 505, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 345, + 306 + ], + "score": 1.0, + "content": "Table 2: For each attention head, we show the pattern from", + "type": "text" + }, + { + "bbox": [ + 345, + 294, + 361, + 304 + ], + "score": 0.81, + "content": "\" = \"", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 293, + 372, + 306 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 373, + 296, + 379, + 303 + ], + "score": 0.69, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 293, + 479, + 306 + ], + "score": 1.0, + "content": "is well approximated by", + "type": "text" + }, + { + "bbox": [ + 480, + 294, + 505, + 304 + ], + "score": 0.83, + "content": "0 . 5 +", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 107, + 304, + 310, + 316 + ], + "score": 0.88, + "content": "\\alpha ( \\cos ( w _ { k } a ) - \\cos ( w _ { k } b ) ) + \\beta ( \\sin ( w _ { k } a ) - \\sin ( w _ { k } b ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "and give the coefficients and fraction of variance", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 316, + 242, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 242, + 328 + ], + "score": 1.0, + "content": "explained for this approximation.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 108, + 352, + 416, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 351, + 417, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 417, + 364 + ], + "score": 1.0, + "content": "C.1.1 PERIODICITY IN THE ACTIVATIONS OF OTHER ATTENTION HEADS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 371, + 505, + 416 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 368, + 383 + ], + "score": 1.0, + "content": "In Figure 9 we plot the attention patterns from the final token", + "type": "text" + }, + { + "bbox": [ + 368, + 372, + 384, + 382 + ], + "score": 0.82, + "content": "\" = \"", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 371, + 459, + 383 + ], + "score": 1.0, + "content": "to the first token", + "type": "text" + }, + { + "bbox": [ + 459, + 374, + 466, + 382 + ], + "score": 0.71, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 371, + 506, + 383 + ], + "score": 1.0, + "content": "for all 4", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 383, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 286, + 395 + ], + "score": 1.0, + "content": "attention heads, as a heatmap over the inputs", + "type": "text" + }, + { + "bbox": [ + 287, + 385, + 294, + 393 + ], + "score": 0.76, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 383, + 311, + 395 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 312, + 384, + 317, + 393 + ], + "score": 0.71, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 383, + 505, + 395 + ], + "score": 1.0, + "content": ", as this is a scalar for each head. We observe a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "score": 1.0, + "content": "striking periodicity and further that heads 1 and 3 represent the same frequency while heads 0 and 2", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 405, + 160, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 160, + 417 + ], + "score": 1.0, + "content": "are different.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 421, + 504, + 444 + ], + "lines": [ + { + "bbox": [ + 105, + 420, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 315, + 435 + ], + "score": 1.0, + "content": "As shown in Appendix A.1, the attention paid from", + "type": "text" + }, + { + "bbox": [ + 315, + 422, + 331, + 432 + ], + "score": 0.8, + "content": "\" = \"", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 420, + 430, + 435 + ], + "score": 1.0, + "content": "to itself is negligible, so", + "type": "text" + }, + { + "bbox": [ + 431, + 420, + 487, + 434 + ], + "score": 0.96, + "content": "A _ { 0 } ^ { j } = 1 - A _ { 1 } ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 420, + 506, + 435 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 432, + 234, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 223, + 445 + ], + "score": 1.0, + "content": "it suffices to plot attention to", + "type": "text" + }, + { + "bbox": [ + 223, + 435, + 230, + 443 + ], + "score": 0.7, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 432, + 234, + 445 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 108, + 460, + 412, + 471 + ], + "lines": [ + { + "bbox": [ + 106, + 459, + 413, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 413, + 473 + ], + "score": 1.0, + "content": "C.1.2 APPROXIMATING ATTENTION HEADS WITH SINES AND COSINES", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 480, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "Attention heads approximately compute degree-2 polynomials of a single frequency or are", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 174, + 505 + ], + "score": 1.0, + "content": "used to amplify", + "type": "text" + }, + { + "bbox": [ + 175, + 492, + 192, + 502 + ], + "score": 0.88, + "content": "W _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 492, + 322, + 505 + ], + "score": 1.0, + "content": ". In order to compute terms like", + "type": "text" + }, + { + "bbox": [ + 323, + 492, + 385, + 504 + ], + "score": 0.89, + "content": "\\cos { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 492, + 505, + 505 + ], + "score": 1.0, + "content": ", the model needs to compute", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 339, + 515 + ], + "score": 1.0, + "content": "the product of the sine and cosine embeddings output by", + "type": "text" + }, + { + "bbox": [ + 340, + 503, + 357, + 514 + ], + "score": 0.89, + "content": "W _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 502, + 505, + 515 + ], + "score": 1.0, + "content": ". As the attention heads are approx-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "score": 1.0, + "content": "imately bilinear (product of attention weights and OV circuit), they are a natural place to perform", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 525, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 506, + 537 + ], + "score": 1.0, + "content": "this computation. Indeed, for each head, the attention scores’ Fourier transform is concentrated on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 536, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 182, + 548 + ], + "score": 1.0, + "content": "a single frequency", + "type": "text" + }, + { + "bbox": [ + 183, + 537, + 196, + 546 + ], + "score": 0.86, + "content": "w _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 536, + 506, + 548 + ], + "score": 1.0, + "content": ". For two of the four heads, the corresponding OV circuit is concentrated on", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "score": 1.0, + "content": "that same frequency. Moreover, the softmax mapping the attention scores to attention weights is in", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "a regime where it behaves approximately linearly (and replacing it with a linear function actually", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 568, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 580 + ], + "score": 1.0, + "content": "improves performance). Thus the attention weights multiply with the OV output to create degree-2", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 579, + 475, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 225, + 592 + ], + "score": 1.0, + "content": "polynomials of the frequency", + "type": "text" + }, + { + "bbox": [ + 226, + 581, + 239, + 590 + ], + "score": 0.86, + "content": "w _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 579, + 475, + 592 + ], + "score": 1.0, + "content": ", as would be needed for the cosine/sine addition formulas.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 596, + 503, + 630 + ], + "lines": [ + { + "bbox": [ + 106, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "For the remaining two heads, their attention scores approximately sum to one and the OV circuits", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "score": 1.0, + "content": "contain all five key frequencies, suggesting that they are used to increase the magnitude of key", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 618, + 447, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 447, + 630 + ], + "score": 1.0, + "content": "frequencies in the residual stream. We confirm all of these claims in Appendix C.1.3.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 106, + 646, + 497, + 667 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 498, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 498, + 657 + ], + "score": 1.0, + "content": "C.1.3 THE ATTENTION PATTERN WEIGHTS ARE WELL APPROXIMATED BY DIFFERENCES OF", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 140, + 658, + 340, + 668 + ], + "spans": [ + { + "bbox": [ + 140, + 658, + 340, + 668 + ], + "score": 1.0, + "content": "SINES AND COSINES OF A SINGLE FREQUENCY.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 675, + 504, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 358, + 690 + ], + "score": 1.0, + "content": "The periodicity of the attention heads has a striking form—", + "type": "text" + }, + { + "bbox": [ + 358, + 675, + 371, + 689 + ], + "score": 0.87, + "content": "A _ { 0 } ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 675, + 478, + 690 + ], + "score": 1.0, + "content": "is well approximated by", + "type": "text" + }, + { + "bbox": [ + 479, + 677, + 504, + 687 + ], + "score": 0.81, + "content": "0 . 5 +", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 686, + 504, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 333, + 700 + ], + "score": 0.86, + "content": "\\alpha ^ { j } ( \\dot { \\cos } ( w _ { k } a ) \\dot { - } \\cos ( w _ { k } b ) ) + \\beta ^ { j } ( \\sin ( w _ { k } a ) - \\sin ( w _ { k } \\bar { b } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 686, + 419, + 701 + ], + "score": 1.0, + "content": ", for some frequency", + "type": "text" + }, + { + "bbox": [ + 419, + 690, + 432, + 699 + ], + "score": 0.84, + "content": "w _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 686, + 492, + 701 + ], + "score": 1.0, + "content": "and constants", + "type": "text" + }, + { + "bbox": [ + 492, + 687, + 504, + 698 + ], + "score": 0.85, + "content": "\\alpha ^ { j }", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 124, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 699, + 135, + 710 + ], + "score": 0.86, + "content": "\\beta ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 698, + 419, + 712 + ], + "score": 1.0, + "content": "(which may differ for each head). 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All four attention heads exhibit", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 206, + 187, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 187, + 219 + ], + "score": 1.0, + "content": "striking periodicity.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 231, + 388, + 286 + ], + "lines": [ + { + "bbox": [ + 221, + 231, + 386, + 286 + ], + "spans": [ + { + "bbox": [ + 221, + 231, + 386, + 286 + ], + "score": 0.43, + "content": "\\begin{array}{c} \\begin{array} { r l r } & { } & { \\frac { \\mathrm { H e a d } } { 0 } \\parallel \\boldsymbol { k } \\parallel \\quad \\begin{array} { c } { \\alpha ^ { j } } \\\\ { 3 5 } \\end{array} | \\quad \\frac { \\alpha ^ { j } } { - 0 . 2 6 } \\parallel \\quad \\begin{array} { c } { \\beta ^ { j } } \\\\ { - 0 . 1 4 } \\end{array} | \\quad \\mathrm { H V E } } \\\\ & { } & { 1 } \\\\ & { } & { 2 } \\\\ & { } & { 3 } \\end{array} | \\begin{array} { l } { 4 2 } \\\\ { 5 2 } \\\\ { 4 2 } \\end{array} | \\quad 0 . 2 7 \\parallel \\quad \\begin{array} { c } { \\beta ^ { j } } \\\\ { 0 . 0 4 } \\end{array} | \\quad 9 9 . 0 3 \\% \\end{array}", + "type": "interline_equation", + "image_path": "265a5ccc87ded124f4a0d94f8dd39157688a00103723e08ef07159bb1c430b34.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 221, + 231, + 388, + 244.75 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 221, + 244.75, + 388, + 258.5 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 221, + 258.5, + 388, + 272.25 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 221, + 272.25, + 388, + 286.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 293, + 505, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 345, + 306 + ], + "score": 1.0, + "content": "Table 2: For each attention head, we show the pattern from", + "type": "text" + }, + { + "bbox": [ + 345, + 294, + 361, + 304 + ], + "score": 0.81, + "content": "\" = \"", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 293, + 372, + 306 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 373, + 296, + 379, + 303 + ], + "score": 0.69, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 293, + 479, + 306 + ], + "score": 1.0, + "content": "is well approximated by", + "type": "text" + }, + { + "bbox": [ + 480, + 294, + 505, + 304 + ], + "score": 0.83, + "content": "0 . 5 +", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 107, + 304, + 310, + 316 + ], + "score": 0.88, + "content": "\\alpha ( \\cos ( w _ { k } a ) - \\cos ( w _ { k } b ) ) + \\beta ( \\sin ( w _ { k } a ) - \\sin ( w _ { k } b ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "and give the coefficients and fraction of variance", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 316, + 242, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 242, + 328 + ], + "score": 1.0, + "content": "explained for this approximation.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 293, + 505, + 328 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 352, + 416, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 351, + 417, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 417, + 364 + ], + "score": 1.0, + "content": "C.1.1 PERIODICITY IN THE ACTIVATIONS OF OTHER ATTENTION HEADS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 371, + 505, + 416 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 368, + 383 + ], + "score": 1.0, + "content": "In Figure 9 we plot the attention patterns from the final token", + "type": "text" + }, + { + "bbox": [ + 368, + 372, + 384, + 382 + ], + "score": 0.82, + "content": "\" = \"", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 371, + 459, + 383 + ], + "score": 1.0, + "content": "to the first token", + "type": "text" + }, + { + "bbox": [ + 459, + 374, + 466, + 382 + ], + "score": 0.71, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 371, + 506, + 383 + ], + "score": 1.0, + "content": "for all 4", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 383, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 286, + 395 + ], + "score": 1.0, + "content": "attention heads, as a heatmap over the inputs", + "type": "text" + }, + { + "bbox": [ + 287, + 385, + 294, + 393 + ], + "score": 0.76, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 383, + 311, + 395 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 312, + 384, + 317, + 393 + ], + "score": 0.71, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 383, + 505, + 395 + ], + "score": 1.0, + "content": ", as this is a scalar for each head. 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In order to compute terms like", + "type": "text" + }, + { + "bbox": [ + 323, + 492, + 385, + 504 + ], + "score": 0.89, + "content": "\\cos { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 492, + 505, + 505 + ], + "score": 1.0, + "content": ", the model needs to compute", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 339, + 515 + ], + "score": 1.0, + "content": "the product of the sine and cosine embeddings output by", + "type": "text" + }, + { + "bbox": [ + 340, + 503, + 357, + 514 + ], + "score": 0.89, + "content": "W _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 502, + 505, + 515 + ], + "score": 1.0, + "content": ". As the attention heads are approx-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "score": 1.0, + "content": "imately bilinear (product of attention weights and OV circuit), they are a natural place to perform", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 525, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 506, + 537 + ], + "score": 1.0, + "content": "this computation. Indeed, for each head, the attention scores’ Fourier transform is concentrated on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 536, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 182, + 548 + ], + "score": 1.0, + "content": "a single frequency", + "type": "text" + }, + { + "bbox": [ + 183, + 537, + 196, + 546 + ], + "score": 0.86, + "content": "w _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 536, + 506, + 548 + ], + "score": 1.0, + "content": ". For two of the four heads, the corresponding OV circuit is concentrated on", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "score": 1.0, + "content": "that same frequency. Moreover, the softmax mapping the attention scores to attention weights is in", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "a regime where it behaves approximately linearly (and replacing it with a linear function actually", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 568, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 580 + ], + "score": 1.0, + "content": "improves performance). 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We confirm all of these claims in Appendix C.1.3.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 596, + 505, + 630 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 646, + 497, + 667 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 498, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 498, + 657 + ], + "score": 1.0, + "content": "C.1.3 THE ATTENTION PATTERN WEIGHTS ARE WELL APPROXIMATED BY DIFFERENCES OF", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 140, + 658, + 340, + 668 + ], + "spans": [ + { + "bbox": [ + 140, + 658, + 340, + 668 + ], + "score": 1.0, + "content": "SINES AND COSINES OF A SINGLE FREQUENCY.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 675, + 504, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 358, + 690 + ], + "score": 1.0, + "content": "The periodicity of the attention heads has a striking form—", + "type": "text" + }, + { + "bbox": [ + 358, + 675, + 371, + 689 + ], + "score": 0.87, + "content": "A _ { 0 } ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 675, + 478, + 690 + ], + "score": 1.0, + "content": "is well approximated by", + "type": "text" + }, + { + "bbox": [ + 479, + 677, + 504, + 687 + ], + "score": 0.81, + "content": "0 . 5 +", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 686, + 504, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 333, + 700 + ], + "score": 0.86, + "content": "\\alpha ^ { j } ( \\dot { \\cos } ( w _ { k } a ) \\dot { - } \\cos ( w _ { k } b ) ) + \\beta ^ { j } ( \\sin ( w _ { k } a ) - \\sin ( w _ { k } \\bar { b } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 686, + 419, + 701 + ], + "score": 1.0, + "content": ", for some frequency", + "type": "text" + }, + { + "bbox": [ + 419, + 690, + 432, + 699 + ], + "score": 0.84, + "content": "w _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 686, + 492, + 701 + ], + "score": 1.0, + "content": "and constants", + "type": "text" + }, + { + "bbox": [ + 492, + 687, + 504, + 698 + ], + "score": 0.85, + "content": "\\alpha ^ { j }", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 124, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 699, + 135, + 710 + ], + "score": 0.86, + "content": "\\beta ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 698, + 419, + 712 + ], + "score": 1.0, + "content": "(which may differ for each head). Note further that this simplifies to", + "type": "text" + }, + { + "bbox": [ + 419, + 699, + 505, + 711 + ], + "score": 0.88, + "content": "0 . 5 + \\gamma ( \\cos ( w _ { k } ( a +", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 194, + 721 + ], + "score": 0.87, + "content": "\\theta ) ) - \\cos ( w _ { k } ( b + \\theta ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 709, + 272, + 722 + ], + "score": 1.0, + "content": "for some constants", + "type": "text" + }, + { + "bbox": [ + 273, + 712, + 280, + 721 + ], + "score": 0.83, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 709, + 297, + 722 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 298, + 710, + 304, + 720 + ], + "score": 0.75, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 709, + 506, + 722 + ], + "score": 1.0, + "content": ". We show the coefficients and fraction of variance", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 720, + 189, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 189, + 732 + ], + "score": 1.0, + "content": "explained in Table 1", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 675, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 116, + 93, + 479, + 305 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 116, + 93, + 479, + 305 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 116, + 93, + 479, + 305 + ], + "spans": [ + { + "bbox": [ + 116, + 93, + 479, + 305 + ], + "score": 0.975, + "type": "image", + "image_path": "83bdc157ca2f28d1861203cdb979e5506c5547ec171ec51bfe348a4587322713.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 116, + 93, + 479, + 163.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 116, + 163.66666666666669, + 479, + 234.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 116, + 234.33333333333337, + 479, + 305.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 318, + 506, + 352 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 317, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 303, + 331 + ], + "score": 1.0, + "content": "Figure 10: We plot the attention pattern weights", + "type": "text" + }, + { + "bbox": [ + 304, + 318, + 316, + 330 + ], + "score": 0.88, + "content": "C _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 317, + 506, + 331 + ], + "score": 1.0, + "content": "in the Fourier basis for each of the four heads", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 107, + 329, + 168, + 341 + ], + "score": 0.92, + "content": "j \\in \\{ 0 , 1 , 2 , 3 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 329, + 505, + 341 + ], + "score": 1.0, + "content": ". We observe significant sparsity, with almost all of each term being associated with", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 339, + 185, + 354 + ], + "spans": [ + { + "bbox": [ + 104, + 339, + 185, + 354 + ], + "score": 1.0, + "content": "a single frequency.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 383, + 503, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 382, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 397 + ], + "score": 1.0, + "content": "Mechanistic Analysis of Attention Patterns. We can further mechanistically analyse how the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 393, + 439, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 439, + 408 + ], + "score": 1.0, + "content": "model achieves this form. The following is a high-level sketch of what is going on:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 411, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 336, + 424 + ], + "score": 1.0, + "content": "First, note that the attention score on position 0 and head", + "type": "text" + }, + { + "bbox": [ + 337, + 412, + 343, + 423 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 410, + 505, + 424 + ], + "score": 1.0, + "content": "is just a lookup table on the input token", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 419, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 113, + 437 + ], + "score": 0.69, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 419, + 148, + 447 + ], + "score": 1.0, + "content": "(of size", + "type": "text" + }, + { + "bbox": [ + 148, + 428, + 157, + 437 + ], + "score": 0.8, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 419, + 257, + 447 + ], + "score": 1.0, + "content": "). To see why, note that", + "type": "text" + }, + { + "bbox": [ + 257, + 423, + 369, + 441 + ], + "score": 0.94, + "content": "A _ { 0 } ^ { j } = m { x _ { 0 } ^ { ( 0 ) } } ^ { T } W _ { K } ^ { j ^ { T } } W _ { Q } ^ { j } x _ { 2 } ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 419, + 375, + 447 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 376, + 424, + 393, + 440 + ], + "score": 0.88, + "content": "x _ { 2 } ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "is constant since the token", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 101, + 440, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 101, + 440, + 147, + 459 + ], + "score": 1.0, + "content": "is always", + "type": "text" + }, + { + "bbox": [ + 147, + 443, + 163, + 454 + ], + "score": 0.77, + "content": "\" = \"", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 440, + 183, + 459 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 183, + 441, + 264, + 455 + ], + "score": 0.92, + "content": "x _ { 0 } ^ { ( 0 ) } = W _ { E } t _ { 0 } + p _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 440, + 348, + 459 + ], + "score": 1.0, + "content": ". So this reduces to", + "type": "text" + }, + { + "bbox": [ + 348, + 443, + 399, + 455 + ], + "score": 0.91, + "content": "t _ { 0 } \\cdot C _ { j } + D", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 440, + 506, + 459 + ], + "score": 1.0, + "content": "for some constant vector", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 451, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 230, + 473 + ], + "score": 0.94, + "content": "C _ { j } = W _ { E } ^ { T } { W _ { K } ^ { j } } ^ { T } W _ { Q } ^ { j } x _ { 2 } ^ { ( 0 ) } \\in \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 451, + 300, + 478 + ], + "score": 1.0, + "content": "and some scalar", + "type": "text" + }, + { + "bbox": [ + 300, + 455, + 392, + 472 + ], + "score": 0.92, + "content": "D = p _ { 0 } ^ { T } { W _ { K } ^ { j } } ^ { T } W _ { Q } ^ { j } x _ { 2 } ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 458, + 412, + 470 + ], + "score": 1.0, + "content": "As", + "type": "text" + }, + { + "bbox": [ + 412, + 459, + 422, + 470 + ], + "score": 0.84, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "is one-hot encoded,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 470, + 365, + 486 + ], + "spans": [ + { + "bbox": [ + 104, + 470, + 342, + 486 + ], + "score": 1.0, + "content": "this is just a lookup table, which we may instead denote as", + "type": "text" + }, + { + "bbox": [ + 342, + 472, + 365, + 484 + ], + "score": 0.87, + "content": "C _ { j } [ a ]", + "type": "inline_equation" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 488, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 487, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 273, + 501 + ], + "score": 1.0, + "content": "Next, note that the attention pattern from", + "type": "text" + }, + { + "bbox": [ + 273, + 489, + 302, + 499 + ], + "score": 0.73, + "content": "= \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 487, + 313, + 501 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 313, + 488, + 383, + 501 + ], + "score": 0.92, + "content": "\\sigma ( C _ { j } [ a ] - C _ { j } [ b ] )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 487, + 505, + 501 + ], + "score": 1.0, + "content": ". As argued in Appendix A.1,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 500, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 177, + 511 + ], + "score": 1.0, + "content": "the attention paid", + "type": "text" + }, + { + "bbox": [ + 177, + 500, + 206, + 510 + ], + "score": 0.85, + "content": "= =", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 500, + 506, + 511 + ], + "score": 1.0, + "content": "is negligible and can be ignored. So the softmax reduces to a softmax over", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 377, + 523 + ], + "score": 1.0, + "content": "two elements, which is a sigmoid on their difference. As form of", + "type": "text" + }, + { + "bbox": [ + 378, + 510, + 390, + 523 + ], + "score": 0.89, + "content": "C _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "does not mention the token", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 521, + 306, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 306, + 533 + ], + "score": 1.0, + "content": "index or value, it is the same for position 0 and 1.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 538, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 181, + 552 + ], + "score": 1.0, + "content": "We now show that", + "type": "text" + }, + { + "bbox": [ + 181, + 539, + 193, + 550 + ], + "score": 0.86, + "content": "C _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 537, + 374, + 552 + ], + "score": 1.0, + "content": "is well-approximated by a wave of frequency", + "type": "text" + }, + { + "bbox": [ + 374, + 540, + 390, + 551 + ], + "score": 0.88, + "content": "w _ { k _ { j } }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 537, + 458, + 552 + ], + "score": 1.0, + "content": "for some integer", + "type": "text" + }, + { + "bbox": [ + 458, + 539, + 469, + 551 + ], + "score": 0.88, + "content": "k _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 537, + 506, + 552 + ], + "score": 1.0, + "content": ". That is,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 548, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 107, + 549, + 261, + 562 + ], + "score": 0.87, + "content": "C _ { j } [ a ] \\approx F _ { j } \\cos ( w _ { k _ { j } } \\tilde { a } ) + G _ { j } \\sin ( w _ { k _ { j } } a )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 548, + 395, + 563 + ], + "score": 1.0, + "content": ". We do this by simply computing", + "type": "text" + }, + { + "bbox": [ + 396, + 550, + 407, + 561 + ], + "score": 0.89, + "content": "C _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 548, + 506, + 563 + ], + "score": 1.0, + "content": "and fitting the constants", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 118, + 572 + ], + "score": 0.86, + "content": "F _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 560, + 136, + 573 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 136, + 561, + 149, + 572 + ], + "score": 0.86, + "content": "G _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 560, + 199, + 573 + ], + "score": 1.0, + "content": "to minimize", + "type": "text" + }, + { + "bbox": [ + 199, + 561, + 209, + 571 + ], + "score": 0.84, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "loss, and display the resulting coefficients for each head in Figure 10. This", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 569, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 149, + 585 + ], + "score": 1.0, + "content": "fit explain", + "type": "text" + }, + { + "bbox": [ + 150, + 571, + 182, + 582 + ], + "score": 0.83, + "content": "9 9 . 0 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 569, + 186, + 585 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 186, + 572, + 218, + 582 + ], + "score": 0.77, + "content": "9 5 . 2 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 569, + 222, + 585 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 222, + 571, + 254, + 582 + ], + "score": 0.8, + "content": "9 9 . 1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 569, + 258, + 585 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 259, + 571, + 291, + 582 + ], + "score": 0.85, + "content": "9 2 . 4 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 569, + 366, + 585 + ], + "score": 1.0, + "content": "of the variance of", + "type": "text" + }, + { + "bbox": [ + 367, + 571, + 379, + 583 + ], + "score": 0.89, + "content": "C _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 569, + 505, + 585 + ], + "score": 1.0, + "content": "respectively. Interestingly, the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 581, + 405, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 405, + 595 + ], + "score": 1.0, + "content": "coefficients of heads 1 and 3 are almost exactly the opposite of each other.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 598, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 597, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 165, + 613 + ], + "score": 1.0, + "content": "For each head", + "type": "text" + }, + { + "bbox": [ + 165, + 600, + 172, + 611 + ], + "score": 0.58, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 597, + 176, + 613 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 176, + 599, + 362, + 612 + ], + "score": 0.89, + "content": "\\sigma ( C _ { j } [ a ] - C _ { j } [ b ] ) \\approx 0 . 5 + E _ { j } ( C _ { j } [ a ] - C _ { j } [ b ] )", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 597, + 439, + 613 + ], + "score": 1.0, + "content": "for some constant", + "type": "text" + }, + { + "bbox": [ + 439, + 600, + 451, + 611 + ], + "score": 0.88, + "content": "E _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 597, + 506, + 613 + ], + "score": 1.0, + "content": "—that is, the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 609, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 624 + ], + "score": 1.0, + "content": "sigmoid has some linear approximation. (The intercept will be 0.5 by symmetry.) The striking", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 620, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 634 + ], + "score": 1.0, + "content": "thing is that, because the inputs to the sigmoid for the attention heads are over a fairly wide range", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 631, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 107, + 632, + 138, + 644 + ], + "score": 0.88, + "content": "( [ - 5 , 5 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 631, + 465, + 645 + ], + "score": 1.0, + "content": "roughly), the linear approximation to the sigmoid is a fairly good fit, explaining", + "type": "text" + }, + { + "bbox": [ + 465, + 632, + 493, + 642 + ], + "score": 0.85, + "content": "9 7 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 631, + 506, + 645 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 642, + 160, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 160, + 655 + ], + "score": 1.0, + "content": "the variance.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 503, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 658, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 505, + 672 + ], + "score": 1.0, + "content": "We validate that this is all that is going on, by replacing the sigmoid with the best linear fit. This", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 670, + 410, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 302, + 682 + ], + "score": 1.0, + "content": "improves performance, decreasing test loss from", + "type": "text" + }, + { + "bbox": [ + 302, + 670, + 348, + 681 + ], + "score": 0.88, + "content": "2 . 4 1 \\cdot 1 \\mathrm { { 0 } } ^ { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 670, + 360, + 682 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 360, + 670, + 406, + 681 + ], + "score": 0.89, + "content": "\\overline { { 2 } } . 1 2 \\cdot 1 0 ^ { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 670, + 410, + 682 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "By properties of sinusoidal functions, the attention patterns of each head will be well approximated", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 118, + 712 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 118, + 699, + 315, + 711 + ], + "score": 0.88, + "content": "0 . 5 \\pm C _ { j } ( \\cos ( w _ { k _ { j } } ( a + \\theta _ { j } ) ) - \\cos ( w _ { k _ { j } } ( b + \\theta _ { j } ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "- the softmax is linear, with an intercept of 0.5,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 171, + 723 + ], + "score": 1.0, + "content": "and the weights", + "type": "text" + }, + { + "bbox": [ + 171, + 711, + 183, + 722 + ], + "score": 0.87, + "content": "C _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "map each token to a score that is a wave in a single frequency. This exactly gives", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 720, + 266, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 266, + 733 + ], + "score": 1.0, + "content": "us the periodic form shown in Figure 9.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5 + } + ], + "page_idx": 15, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "16", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 116, + 93, + 479, + 305 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 116, + 93, + 479, + 305 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 116, + 93, + 479, + 305 + ], + "spans": [ + { + "bbox": [ + 116, + 93, + 479, + 305 + ], + "score": 0.975, + "type": "image", + "image_path": "83bdc157ca2f28d1861203cdb979e5506c5547ec171ec51bfe348a4587322713.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 116, + 93, + 479, + 163.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 116, + 163.66666666666669, + 479, + 234.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 116, + 234.33333333333337, + 479, + 305.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 318, + 506, + 352 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 317, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 303, + 331 + ], + "score": 1.0, + "content": "Figure 10: We plot the attention pattern weights", + "type": "text" + }, + { + "bbox": [ + 304, + 318, + 316, + 330 + ], + "score": 0.88, + "content": "C _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 317, + 506, + 331 + ], + "score": 1.0, + "content": "in the Fourier basis for each of the four heads", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 107, + 329, + 168, + 341 + ], + "score": 0.92, + "content": "j \\in \\{ 0 , 1 , 2 , 3 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 329, + 505, + 341 + ], + "score": 1.0, + "content": ". We observe significant sparsity, with almost all of each term being associated with", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 339, + 185, + 354 + ], + "spans": [ + { + "bbox": [ + 104, + 339, + 185, + 354 + ], + "score": 1.0, + "content": "a single frequency.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 383, + 503, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 382, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 397 + ], + "score": 1.0, + "content": "Mechanistic Analysis of Attention Patterns. We can further mechanistically analyse how the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 393, + 439, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 439, + 408 + ], + "score": 1.0, + "content": "model achieves this form. The following is a high-level sketch of what is going on:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 382, + 505, + 408 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 411, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 336, + 424 + ], + "score": 1.0, + "content": "First, note that the attention score on position 0 and head", + "type": "text" + }, + { + "bbox": [ + 337, + 412, + 343, + 423 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 410, + 505, + 424 + ], + "score": 1.0, + "content": "is just a lookup table on the input token", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 419, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 113, + 437 + ], + "score": 0.69, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 419, + 148, + 447 + ], + "score": 1.0, + "content": "(of size", + "type": "text" + }, + { + "bbox": [ + 148, + 428, + 157, + 437 + ], + "score": 0.8, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 419, + 257, + 447 + ], + "score": 1.0, + "content": "). To see why, note that", + "type": "text" + }, + { + "bbox": [ + 257, + 423, + 369, + 441 + ], + "score": 0.94, + "content": "A _ { 0 } ^ { j } = m { x _ { 0 } ^ { ( 0 ) } } ^ { T } W _ { K } ^ { j ^ { T } } W _ { Q } ^ { j } x _ { 2 } ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 419, + 375, + 447 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 376, + 424, + 393, + 440 + ], + "score": 0.88, + "content": "x _ { 2 } ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "is constant since the token", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 101, + 440, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 101, + 440, + 147, + 459 + ], + "score": 1.0, + "content": "is always", + "type": "text" + }, + { + "bbox": [ + 147, + 443, + 163, + 454 + ], + "score": 0.77, + "content": "\" = \"", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 440, + 183, + 459 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 183, + 441, + 264, + 455 + ], + "score": 0.92, + "content": "x _ { 0 } ^ { ( 0 ) } = W _ { E } t _ { 0 } + p _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 440, + 348, + 459 + ], + "score": 1.0, + "content": ". So this reduces to", + "type": "text" + }, + { + "bbox": [ + 348, + 443, + 399, + 455 + ], + "score": 0.91, + "content": "t _ { 0 } \\cdot C _ { j } + D", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 440, + 506, + 459 + ], + "score": 1.0, + "content": "for some constant vector", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 451, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 230, + 473 + ], + "score": 0.94, + "content": "C _ { j } = W _ { E } ^ { T } { W _ { K } ^ { j } } ^ { T } W _ { Q } ^ { j } x _ { 2 } ^ { ( 0 ) } \\in \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 451, + 300, + 478 + ], + "score": 1.0, + "content": "and some scalar", + "type": "text" + }, + { + "bbox": [ + 300, + 455, + 392, + 472 + ], + "score": 0.92, + "content": "D = p _ { 0 } ^ { T } { W _ { K } ^ { j } } ^ { T } W _ { Q } ^ { j } x _ { 2 } ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 458, + 412, + 470 + ], + "score": 1.0, + "content": "As", + "type": "text" + }, + { + "bbox": [ + 412, + 459, + 422, + 470 + ], + "score": 0.84, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "is one-hot encoded,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 470, + 365, + 486 + ], + "spans": [ + { + "bbox": [ + 104, + 470, + 342, + 486 + ], + "score": 1.0, + "content": "this is just a lookup table, which we may instead denote as", + "type": "text" + }, + { + "bbox": [ + 342, + 472, + 365, + 484 + ], + "score": 0.87, + "content": "C _ { j } [ a ]", + "type": "inline_equation" + } + ], + "index": 12 + } + ], + "index": 10, + "bbox_fs": [ + 101, + 410, + 506, + 486 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 488, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 487, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 273, + 501 + ], + "score": 1.0, + "content": "Next, note that the attention pattern from", + "type": "text" + }, + { + "bbox": [ + 273, + 489, + 302, + 499 + ], + "score": 0.73, + "content": "= \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 487, + 313, + 501 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 313, + 488, + 383, + 501 + ], + "score": 0.92, + "content": "\\sigma ( C _ { j } [ a ] - C _ { j } [ b ] )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 487, + 505, + 501 + ], + "score": 1.0, + "content": ". As argued in Appendix A.1,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 500, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 177, + 511 + ], + "score": 1.0, + "content": "the attention paid", + "type": "text" + }, + { + "bbox": [ + 177, + 500, + 206, + 510 + ], + "score": 0.85, + "content": "= =", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 500, + 506, + 511 + ], + "score": 1.0, + "content": "is negligible and can be ignored. So the softmax reduces to a softmax over", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 377, + 523 + ], + "score": 1.0, + "content": "two elements, which is a sigmoid on their difference. As form of", + "type": "text" + }, + { + "bbox": [ + 378, + 510, + 390, + 523 + ], + "score": 0.89, + "content": "C _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "does not mention the token", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 521, + 306, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 306, + 533 + ], + "score": 1.0, + "content": "index or value, it is the same for position 0 and 1.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 487, + 506, + 533 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 538, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 181, + 552 + ], + "score": 1.0, + "content": "We now show that", + "type": "text" + }, + { + "bbox": [ + 181, + 539, + 193, + 550 + ], + "score": 0.86, + "content": "C _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 537, + 374, + 552 + ], + "score": 1.0, + "content": "is well-approximated by a wave of frequency", + "type": "text" + }, + { + "bbox": [ + 374, + 540, + 390, + 551 + ], + "score": 0.88, + "content": "w _ { k _ { j } }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 537, + 458, + 552 + ], + "score": 1.0, + "content": "for some integer", + "type": "text" + }, + { + "bbox": [ + 458, + 539, + 469, + 551 + ], + "score": 0.88, + "content": "k _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 537, + 506, + 552 + ], + "score": 1.0, + "content": ". That is,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 548, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 107, + 549, + 261, + 562 + ], + "score": 0.87, + "content": "C _ { j } [ a ] \\approx F _ { j } \\cos ( w _ { k _ { j } } \\tilde { a } ) + G _ { j } \\sin ( w _ { k _ { j } } a )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 548, + 395, + 563 + ], + "score": 1.0, + "content": ". We do this by simply computing", + "type": "text" + }, + { + "bbox": [ + 396, + 550, + 407, + 561 + ], + "score": 0.89, + "content": "C _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 548, + 506, + 563 + ], + "score": 1.0, + "content": "and fitting the constants", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 118, + 572 + ], + "score": 0.86, + "content": "F _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 560, + 136, + 573 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 136, + 561, + 149, + 572 + ], + "score": 0.86, + "content": "G _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 560, + 199, + 573 + ], + "score": 1.0, + "content": "to minimize", + "type": "text" + }, + { + "bbox": [ + 199, + 561, + 209, + 571 + ], + "score": 0.84, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "loss, and display the resulting coefficients for each head in Figure 10. This", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 569, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 149, + 585 + ], + "score": 1.0, + "content": "fit explain", + "type": "text" + }, + { + "bbox": [ + 150, + 571, + 182, + 582 + ], + "score": 0.83, + "content": "9 9 . 0 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 569, + 186, + 585 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 186, + 572, + 218, + 582 + ], + "score": 0.77, + "content": "9 5 . 2 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 569, + 222, + 585 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 222, + 571, + 254, + 582 + ], + "score": 0.8, + "content": "9 9 . 1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 569, + 258, + 585 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 259, + 571, + 291, + 582 + ], + "score": 0.85, + "content": "9 2 . 4 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 569, + 366, + 585 + ], + "score": 1.0, + "content": "of the variance of", + "type": "text" + }, + { + "bbox": [ + 367, + 571, + 379, + 583 + ], + "score": 0.89, + "content": "C _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 569, + 505, + 585 + ], + "score": 1.0, + "content": "respectively. Interestingly, the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 581, + 405, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 405, + 595 + ], + "score": 1.0, + "content": "coefficients of heads 1 and 3 are almost exactly the opposite of each other.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19, + "bbox_fs": [ + 104, + 537, + 506, + 595 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 598, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 597, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 165, + 613 + ], + "score": 1.0, + "content": "For each head", + "type": "text" + }, + { + "bbox": [ + 165, + 600, + 172, + 611 + ], + "score": 0.58, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 597, + 176, + 613 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 176, + 599, + 362, + 612 + ], + "score": 0.89, + "content": "\\sigma ( C _ { j } [ a ] - C _ { j } [ b ] ) \\approx 0 . 5 + E _ { j } ( C _ { j } [ a ] - C _ { j } [ b ] )", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 597, + 439, + 613 + ], + "score": 1.0, + "content": "for some constant", + "type": "text" + }, + { + "bbox": [ + 439, + 600, + 451, + 611 + ], + "score": 0.88, + "content": "E _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 597, + 506, + 613 + ], + "score": 1.0, + "content": "—that is, the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 609, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 624 + ], + "score": 1.0, + "content": "sigmoid has some linear approximation. (The intercept will be 0.5 by symmetry.) The striking", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 620, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 634 + ], + "score": 1.0, + "content": "thing is that, because the inputs to the sigmoid for the attention heads are over a fairly wide range", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 631, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 107, + 632, + 138, + 644 + ], + "score": 0.88, + "content": "( [ - 5 , 5 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 631, + 465, + 645 + ], + "score": 1.0, + "content": "roughly), the linear approximation to the sigmoid is a fairly good fit, explaining", + "type": "text" + }, + { + "bbox": [ + 465, + 632, + 493, + 642 + ], + "score": 0.85, + "content": "9 7 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 631, + 506, + 645 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 642, + 160, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 160, + 655 + ], + "score": 1.0, + "content": "the variance.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 597, + 506, + 655 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 503, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 658, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 505, + 672 + ], + "score": 1.0, + "content": "We validate that this is all that is going on, by replacing the sigmoid with the best linear fit. This", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 670, + 410, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 302, + 682 + ], + "score": 1.0, + "content": "improves performance, decreasing test loss from", + "type": "text" + }, + { + "bbox": [ + 302, + 670, + 348, + 681 + ], + "score": 0.88, + "content": "2 . 4 1 \\cdot 1 \\mathrm { { 0 } } ^ { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 670, + 360, + 682 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 360, + 670, + 406, + 681 + ], + "score": 0.89, + "content": "\\overline { { 2 } } . 1 2 \\cdot 1 0 ^ { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 670, + 410, + 682 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 658, + 505, + 682 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "By properties of sinusoidal functions, the attention patterns of each head will be well approximated", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 118, + 712 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 118, + 699, + 315, + 711 + ], + "score": 0.88, + "content": "0 . 5 \\pm C _ { j } ( \\cos ( w _ { k _ { j } } ( a + \\theta _ { j } ) ) - \\cos ( w _ { k _ { j } } ( b + \\theta _ { j } ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "- the softmax is linear, with an intercept of 0.5,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 171, + 723 + ], + "score": 1.0, + "content": "and the weights", + "type": "text" + }, + { + "bbox": [ + 171, + 711, + 183, + 722 + ], + "score": 0.87, + "content": "C _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "map each token to a score that is a wave in a single frequency. This exactly gives", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 720, + 266, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 266, + 733 + ], + "score": 1.0, + "content": "us the periodic form shown in Figure 9.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 687, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 115, + 93, + 479, + 305 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 115, + 93, + 479, + 305 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 93, + 479, + 305 + ], + "spans": [ + { + "bbox": [ + 115, + 93, + 479, + 305 + ], + "score": 0.969, + "type": "image", + "image_path": "b6a39ebda0305cf2bd0f512c8c52a49588309ebbb12e6fbb115be0284d8beb2f.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 115, + 93, + 479, + 163.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 115, + 163.66666666666669, + 479, + 234.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 115, + 234.33333333333337, + 479, + 305.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 318, + 505, + 397 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 317, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 307, + 333 + ], + "score": 1.0, + "content": "Figure 11: We plot the output of the OV circuit", + "type": "text" + }, + { + "bbox": [ + 307, + 317, + 357, + 332 + ], + "score": 0.93, + "content": "W _ { O } ^ { j } W _ { V } ^ { j } x ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 317, + 506, + 333 + ], + "score": 1.0, + "content": "in the Fourier basis for each of the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 330, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 151, + 344 + ], + "score": 1.0, + "content": "four heads", + "type": "text" + }, + { + "bbox": [ + 152, + 330, + 214, + 343 + ], + "score": 0.94, + "content": "j \\in \\{ 0 , 1 , 2 , 3 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 330, + 370, + 344 + ], + "score": 1.0, + "content": ". As with the attention pattern weights", + "type": "text" + }, + { + "bbox": [ + 371, + 331, + 383, + 343 + ], + "score": 0.89, + "content": "C _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 330, + 506, + 344 + ], + "score": 1.0, + "content": "in Figure 10, we observe that", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 340, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 354 + ], + "score": 1.0, + "content": "the only components with significant norm are those corresponding to key frequencies, and that the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 353, + 504, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 504, + 364 + ], + "score": 1.0, + "content": "largest component corresponds to the frequencies of the attention patterns of the attention heads.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "As attention pattern of heads 1 and 3 are sum to one, but their OV circuits are almost exactly the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 375, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 386 + ], + "score": 1.0, + "content": "same and consist of all five key frequencies, this implies that heads 1 and 3 are used to increase the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 385, + 381, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 381, + 398 + ], + "score": 1.0, + "content": "magnitude of key frequencies in the residual stream (Section C.1.3).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 417, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 104, + 416, + 507, + 433 + ], + "spans": [ + { + "bbox": [ + 104, + 416, + 197, + 433 + ], + "score": 1.0, + "content": "Finally, for each head", + "type": "text" + }, + { + "bbox": [ + 197, + 420, + 203, + 430 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 416, + 355, + 433 + ], + "score": 1.0, + "content": ", we plot the output of the OV circuit", + "type": "text" + }, + { + "bbox": [ + 356, + 417, + 406, + 431 + ], + "score": 0.94, + "content": "W _ { O } ^ { j } W _ { V } ^ { j } x ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 416, + 507, + 433 + ], + "score": 1.0, + "content": "in the Fourier basis and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 429, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 444 + ], + "score": 1.0, + "content": "display the results in Figure 11). The largest component of each head corresponding to the frequency", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 198, + 453 + ], + "score": 1.0, + "content": "of the attention pattern", + "type": "text" + }, + { + "bbox": [ + 198, + 441, + 210, + 453 + ], + "score": 0.89, + "content": "C _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 441, + 505, + 453 + ], + "score": 1.0, + "content": ", with heads 0 and 2 being almost entirely composed of a sines and cosines", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "of a single frequency. On the other hand, the norms for the components of heads 1 and 3 are almost", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "exactly the same, and contain all five key frequencies. As the coefficients of the attention pattern", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 473, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 506, + 487 + ], + "score": 1.0, + "content": "weights have the opposite non-constant components (Table 2, Figure 10), their attention scores sum", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 485, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 506, + 496 + ], + "score": 1.0, + "content": "almost exactly to 1 across all inputs. This implies that heads 1 and 3 are used to output the first", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 157, + 508 + ], + "score": 1.0, + "content": "order terms", + "type": "text" + }, + { + "bbox": [ + 158, + 495, + 233, + 507 + ], + "score": 0.89, + "content": "\\dot { \\sin \\left( w _ { k } \\right) } , \\cos \\left( w _ { k } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 495, + 506, + 508 + ], + "score": 1.0, + "content": "in the five key frequencies. We speculate that this is because of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 280, + 519 + ], + "score": 1.0, + "content": "weight decay encouraging the embeddings", + "type": "text" + }, + { + "bbox": [ + 280, + 507, + 298, + 518 + ], + "score": 0.91, + "content": "W _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "to be small, causing the network to allocate two of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 517, + 344, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 322, + 531 + ], + "score": 1.0, + "content": "its attention heads to effectively increasing the size of", + "type": "text" + }, + { + "bbox": [ + 323, + 518, + 340, + 528 + ], + "score": 0.9, + "content": "W _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 517, + 344, + 531 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 547 + ], + "score": 1.0, + "content": "Bringing it all together, this implies that attention heads 0 and 2 are approximately computing a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 545, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 559 + ], + "score": 1.0, + "content": "degree 2 polynomial of cosines and sines of a single frequency each, while heads 1 and 3 amplify", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 556, + 277, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 277, + 568 + ], + "score": 1.0, + "content": "the key frequencies in the residual stream.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 109, + 581, + 402, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 580, + 403, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 403, + 593 + ], + "score": 1.0, + "content": "C.1.4 PERIODICITY IN THE ACTIVATIONS OF ADDITIONAL NEURONS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 504, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "In Figure 12, we display the activations of four more MLP neurons, as a function of the inputs. As", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 610, + 417, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 417, + 623 + ], + "score": 1.0, + "content": "with neuron 0, the activations of these neurons are also periodic in the inputs.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 106, + 635, + 371, + 645 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 372, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 372, + 647 + ], + "score": 1.0, + "content": "C.1.5 ADDITIONAL GROKKING FIGURES FOR MAINLINE RUN", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 653, + 504, + 676 + ], + "lines": [ + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "score": 1.0, + "content": "In Figure 13, we display the accuracy of the model when restricting the model to use only the five", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 664, + 469, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 469, + 678 + ], + "score": 1.0, + "content": "key frequencies. As with restricted loss, this improves model performance during training.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 681, + 503, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "score": 1.0, + "content": "In Figure 14, we show the coefficients of the five key frequencies in the logits, calculated by regress-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 692, + 334, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 224, + 707 + ], + "score": 1.0, + "content": "ing the logits against the five", + "type": "text" + }, + { + "bbox": [ + 224, + 693, + 305, + 705 + ], + "score": 0.83, + "content": "\\cos { \\left( w _ { k } ( a + b - c ) \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 692, + 334, + 707 + ], + "score": 1.0, + "content": "terms.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "In Figure 15, we plot the excluded loss if we exclude each of the five key frequencies (as opposed to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 720, + 207, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 207, + 734 + ], + "score": 1.0, + "content": "all five key frequencies).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "17", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 115, + 93, + 479, + 305 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 115, + 93, + 479, + 305 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 93, + 479, + 305 + ], + "spans": [ + { + "bbox": [ + 115, + 93, + 479, + 305 + ], + "score": 0.969, + "type": "image", + "image_path": "b6a39ebda0305cf2bd0f512c8c52a49588309ebbb12e6fbb115be0284d8beb2f.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 115, + 93, + 479, + 163.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 115, + 163.66666666666669, + 479, + 234.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 115, + 234.33333333333337, + 479, + 305.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 318, + 505, + 397 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 317, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 307, + 333 + ], + "score": 1.0, + "content": "Figure 11: We plot the output of the OV circuit", + "type": "text" + }, + { + "bbox": [ + 307, + 317, + 357, + 332 + ], + "score": 0.93, + "content": "W _ { O } ^ { j } W _ { V } ^ { j } x ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 317, + 506, + 333 + ], + "score": 1.0, + "content": "in the Fourier basis for each of the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 330, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 151, + 344 + ], + "score": 1.0, + "content": "four heads", + "type": "text" + }, + { + "bbox": [ + 152, + 330, + 214, + 343 + ], + "score": 0.94, + "content": "j \\in \\{ 0 , 1 , 2 , 3 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 330, + 370, + 344 + ], + "score": 1.0, + "content": ". As with the attention pattern weights", + "type": "text" + }, + { + "bbox": [ + 371, + 331, + 383, + 343 + ], + "score": 0.89, + "content": "C _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 330, + 506, + 344 + ], + "score": 1.0, + "content": "in Figure 10, we observe that", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 340, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 354 + ], + "score": 1.0, + "content": "the only components with significant norm are those corresponding to key frequencies, and that the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 353, + 504, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 504, + 364 + ], + "score": 1.0, + "content": "largest component corresponds to the frequencies of the attention patterns of the attention heads.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "As attention pattern of heads 1 and 3 are sum to one, but their OV circuits are almost exactly the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 375, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 386 + ], + "score": 1.0, + "content": "same and consist of all five key frequencies, this implies that heads 1 and 3 are used to increase the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 385, + 381, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 381, + 398 + ], + "score": 1.0, + "content": "magnitude of key frequencies in the residual stream (Section C.1.3).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 417, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 104, + 416, + 507, + 433 + ], + "spans": [ + { + "bbox": [ + 104, + 416, + 197, + 433 + ], + "score": 1.0, + "content": "Finally, for each head", + "type": "text" + }, + { + "bbox": [ + 197, + 420, + 203, + 430 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 416, + 355, + 433 + ], + "score": 1.0, + "content": ", we plot the output of the OV circuit", + "type": "text" + }, + { + "bbox": [ + 356, + 417, + 406, + 431 + ], + "score": 0.94, + "content": "W _ { O } ^ { j } W _ { V } ^ { j } x ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 416, + 507, + 433 + ], + "score": 1.0, + "content": "in the Fourier basis and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 429, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 444 + ], + "score": 1.0, + "content": "display the results in Figure 11). The largest component of each head corresponding to the frequency", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 198, + 453 + ], + "score": 1.0, + "content": "of the attention pattern", + "type": "text" + }, + { + "bbox": [ + 198, + 441, + 210, + 453 + ], + "score": 0.89, + "content": "C _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 441, + 505, + 453 + ], + "score": 1.0, + "content": ", with heads 0 and 2 being almost entirely composed of a sines and cosines", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "of a single frequency. 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WL Component Fourier components of u MLP(a,b) or uT MLP(a,b) FVEcos(w2c) 147.4cos (w2a) cos (w2b)-145.8sin(w2a)sin (w2b) ~ 146.6cos (w2(a +b)) 99.2%sin(w2c) 145.5 cos (wza) sin (w2b) + 145.6 sin(w2a) cos (w2b) ≈ 145.5 sin (w2(a + b)) 99.1%cos (wgc) 49.3cos(wga) cos(wgb) - 48.0 sin (wga) sin (wgb) ~ 48.6 cos (wg(a + b)) 96.4%sin (wgc) 48.6 cos (wga) sin(wgb) + 48.5 sin(wga) cos (wgb) ~ 48.5 sin(wg(a + b)) 96.7%
WL ComponentFourier components of u MLP(a,b) or uT MLP(a,b)oruMLP(a,b)
COSW2C147.4cosw2b)≈146.6cosw2(a+b))
sinW2C145.5 cosw2a) sin(w2b)+145.6sin(w2a) cos(w2b)~145.5sinw2(a+b)）
COSwgC49.3cos49.3cos(wga) cos(wgb) - 48.0 sin (wga) sin (wgb) ~ 48.6 cos (wg(a + b))48.6 cos (wga) sin(wgb) + 48.5 sin(wga) cos (wgb) ~ 48.5 sin(wg(a + b))
sinwgC48.6cosgb)≈48.5sin(wg(a+b))
COSW19c)58.0cos (wiga) cos (w1gb)- 58.3 sin (wiga) sin(w1gb) ~ 58.2 cos(w19(a + b))59.3 cos (w1ga) sin (w1gb) + 59.4 sin (w19a) cos (w19b) ~ 59.4sin(w1g(a + b))94.4cos(w31a) cos (w31b) - 96.4sin (w31a) sin(w3ib) ~ 95.4 cos (w31(a + b))97.2 cos (w31a) sin(w31b) + 97.1 sin (w31a) cos (w3ib) ~ 97.2 sin (w31(a +b)58.0 cos(w
sin(w19c）
sin (w31c)cos (w31c)sin (w31c)
WL Component(a) Seed 1
Fourier components of uTMLP(a, b) or uTMLP(a, b)
COSW40C97.0cOsw40aCOs(w40b)-99.4sin(w4oasinW40b) ~ 98.2 cos (w40(a + b))97.3%92.7%
sinW40C81.3cosw40asinw40b+81.3sin(w4oa)COSw40b)≈ 81.3sin(w4o(a+b))
COSW44C309.1cos(w44a)COsW44b-338.7 sin(w44asins (w44a) sin(w44b) + 327.2 sin (w44a) cos (w44b) ~ 327.3 sin (w44(a + b))））
sinW44C327.3cosW44asin (W44b)+ 327.2 sinw44aCos
COSW53C192.1cos (W53a)COSW53b-192.2 sin(w53asin97.3%95.7%FVE
sinW53C166.7cos(w53asin (W53b+ 166.8 sinw53aCos
WL Component(b) Seed 2
WL ComponentFouriercomponents of uMLPa，borUjMLP(a,b)
COsW31C156.1cos(w31a)cosw31b-156.5 sin(w31asin(w31b~156.3ccs(w31(a+b))99.3%98.9%
sinW31C150.7cos(w31asin (w31b)+ 150.7sinw31aCos(w31b~150.7sin(w31(a+b))
COSW45C72.5 cosW45aW45b）-76.8sinW45asinW45b~ 74.6cos（W45(a+b))95.9%
sinW45C74.7cOsw45asinw45b+74.6sin(w45a)COSw45b~ 74.6sinw45(a+b))96.6%
cos(W49C)45.9 cosw49acosw49b）-45.5sinW49a)sinw49b~45.7cos（w49(a+b))97.0%
sinW49C45.8 cosw49asinw49b+45.8sin(w49a)COsW4gb~ 45.8sinw49(a+b))96.9%
cos((W52C)sin (w52C)71.6cosw52acos(w52b）-72.1sin(W52qsin (W52b~ 71.9 cosw52(a+b))w52(a +b)
68.7cosW52asinW52b）+68.7sin(W52a)CosW52b~ 68.7sin
(c) Seed 3FVE
WLCWL ComponentFouriercomponentsof uMLP(a，b)orvMLP(a,b)
cossin(w17C)cos (w32c)sin(w32c)cos (W42C)(W17C)66.0cosw17acosw17b）-63.5sin(sin(w17b~64.8cosw17(a+b))96.4%94.9%96.2%
W17C66.4cosw17asinw17b）+66.4sin(w17a)Cosw17b~ 66.4sinw1z(a+b))
W32C68.7cosW32acos（w32b-68.4sin(Wa2q)sin (.68.5.cosWaa(a +b))
W32C68.0 cosw32asinW32b+68.0sin(W32a)CosW32b~ 68.0sinw32(a+b))96.3%
W42C100.4cos(w420)cos(w42)-96.0sin(w42asinw42b~98.2cos(w42(a+b))97.9%
W42C100.2 cos (W42a)sin (w42b)+ 100.1 sin(w42gcos(w42b~100.1sii(w42(a+b))98.6%
COSW51C118.0 cos (w51a)CosW51b-116.2 sin(w51asinW51b≈117.1cos(w51(a+b))99.0%
sinW51C114.3cos (w51a)sin (w51b)+ 114.2 sinw51acos(w51b)≈114.2sin(w51(a+b))98.5%
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WL Component Fourier components of u MLP(a,b) or uT MLP(a,b) FVEcos(w2c) 147.4cos (w2a) cos (w2b)-145.8sin(w2a)sin (w2b) ~ 146.6cos (w2(a +b)) 99.2%sin(w2c) 145.5 cos (wza) sin (w2b) + 145.6 sin(w2a) cos (w2b) ≈ 145.5 sin (w2(a + b)) 99.1%cos (wgc) 49.3cos(wga) cos(wgb) - 48.0 sin (wga) sin (wgb) ~ 48.6 cos (wg(a + b)) 96.4%sin (wgc) 48.6 cos (wga) sin(wgb) + 48.5 sin(wga) cos (wgb) ~ 48.5 sin(wg(a + b)) 96.7%
WL ComponentFourier components of u MLP(a,b) or uT MLP(a,b)oruMLP(a,b)
COSW2C147.4cosw2b)≈146.6cosw2(a+b))
sinW2C145.5 cosw2a) sin(w2b)+145.6sin(w2a) cos(w2b)~145.5sinw2(a+b)）
COSwgC49.3cos49.3cos(wga) cos(wgb) - 48.0 sin (wga) sin (wgb) ~ 48.6 cos (wg(a + b))48.6 cos (wga) sin(wgb) + 48.5 sin(wga) cos (wgb) ~ 48.5 sin(wg(a + b))
sinwgC48.6cosgb)≈48.5sin(wg(a+b))
COSW19c)58.0cos (wiga) cos (w1gb)- 58.3 sin (wiga) sin(w1gb) ~ 58.2 cos(w19(a + b))59.3 cos (w1ga) sin (w1gb) + 59.4 sin (w19a) cos (w19b) ~ 59.4sin(w1g(a + b))94.4cos(w31a) cos (w31b) - 96.4sin (w31a) sin(w3ib) ~ 95.4 cos (w31(a + b))97.2 cos (w31a) sin(w31b) + 97.1 sin (w31a) cos (w3ib) ~ 97.2 sin (w31(a +b)58.0 cos(w
sin(w19c）
sin (w31c)cos (w31c)sin (w31c)
WL Component(a) Seed 1
Fourier components of uTMLP(a, b) or uTMLP(a, b)
COSW40C97.0cOsw40aCOs(w40b)-99.4sin(w4oasinW40b) ~ 98.2 cos (w40(a + b))97.3%92.7%
sinW40C81.3cosw40asinw40b+81.3sin(w4oa)COSw40b)≈ 81.3sin(w4o(a+b))
COSW44C309.1cos(w44a)COsW44b-338.7 sin(w44asins (w44a) sin(w44b) + 327.2 sin (w44a) cos (w44b) ~ 327.3 sin (w44(a + b))））
sinW44C327.3cosW44asin (W44b)+ 327.2 sinw44aCos
COSW53C192.1cos (W53a)COSW53b-192.2 sin(w53asin97.3%95.7%FVE
sinW53C166.7cos(w53asin (W53b+ 166.8 sinw53aCos
WL Component(b) Seed 2
WL ComponentFouriercomponents of uMLPa，borUjMLP(a,b)
COsW31C156.1cos(w31a)cosw31b-156.5 sin(w31asin(w31b~156.3ccs(w31(a+b))99.3%98.9%
sinW31C150.7cos(w31asin (w31b)+ 150.7sinw31aCos(w31b~150.7sin(w31(a+b))
COSW45C72.5 cosW45aW45b）-76.8sinW45asinW45b~ 74.6cos（W45(a+b))95.9%
sinW45C74.7cOsw45asinw45b+74.6sin(w45a)COSw45b~ 74.6sinw45(a+b))96.6%
cos(W49C)45.9 cosw49acosw49b）-45.5sinW49a)sinw49b~45.7cos（w49(a+b))97.0%
sinW49C45.8 cosw49asinw49b+45.8sin(w49a)COsW4gb~ 45.8sinw49(a+b))96.9%
cos((W52C)sin (w52C)71.6cosw52acos(w52b）-72.1sin(W52qsin (W52b~ 71.9 cosw52(a+b))w52(a +b)
68.7cosW52asinW52b）+68.7sin(W52a)CosW52b~ 68.7sin
(c) Seed 3FVE
WLCWL ComponentFouriercomponentsof uMLP(a，b)orvMLP(a,b)
cossin(w17C)cos (w32c)sin(w32c)cos (W42C)(W17C)66.0cosw17acosw17b）-63.5sin(sin(w17b~64.8cosw17(a+b))96.4%94.9%96.2%
W17C66.4cosw17asinw17b）+66.4sin(w17a)Cosw17b~ 66.4sinw1z(a+b))
W32C68.7cosW32acos（w32b-68.4sin(Wa2q)sin (.68.5.cosWaa(a +b))
W32C68.0 cosw32asinW32b+68.0sin(W32a)CosW32b~ 68.0sinw32(a+b))96.3%
W42C100.4cos(w420)cos(w42)-96.0sin(w42asinw42b~98.2cos(w42(a+b))97.9%
W42C100.2 cos (W42a)sin (w42b)+ 100.1 sin(w42gcos(w42b~100.1sii(w42(a+b))98.6%
COSW51C118.0 cos (w51a)CosW51b-116.2 sin(w51asinW51b≈117.1cos(w51(a+b))99.0%
sinW51C114.3cos (w51a)sin (w51b)+ 114.2 sinw51acos(w51b)≈114.2sin(w51(a+b))98.5%
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For brevity, we omit terms with coefficients less than", + "type": "text" + }, + { + "bbox": [ + 366, + 604, + 386, + 614 + ], + "score": 0.89, + "content": "1 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 604, + 505, + 615 + ], + "score": 1.0, + "content": "of the largest coefficient. We", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 614, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 505, + 627 + ], + "score": 1.0, + "content": "then compute the fraction of variance explained (FVE) if we replace the projection with a multiple", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 625, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 225, + 638 + ], + "score": 1.0, + "content": "of a single term of the form", + "type": "text" + }, + { + "bbox": [ + 225, + 625, + 289, + 637 + ], + "score": 0.9, + "content": "\\cos { ( w _ { k } ( a \\bar { + } b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 625, + 303, + 638 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 304, + 626, + 366, + 637 + ], + "score": 0.92, + "content": "\\sin { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 625, + 505, + 638 + ], + "score": 1.0, + "content": ", and find that this is consistently", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 636, + 150, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 150, + 648 + ], + "score": 1.0, + "content": "close to 1.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 581, + 506, + 648 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 115, + 94, + 497, + 306 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 115, + 94, + 497, + 306 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 94, + 497, + 306 + ], + "spans": [ + { + "bbox": [ + 115, + 94, + 497, + 306 + ], + "score": 0.976, + "type": "image", + "image_path": "2049e7e629e850219ced239cd20167da1e3876d838ebe35a707f31334989eb9c.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 115, + 94, + 497, + 164.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 115, + 164.66666666666669, + 497, + 235.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 115, + 235.33333333333337, + 497, + 306.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 319, + 505, + 353 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "score": 1.0, + "content": "Figure 17: The norms of the direction corresponding to sine and cosine waves in the neuron-logit", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 158, + 342 + ], + "score": 1.0, + "content": "map weights", + "type": "text" + }, + { + "bbox": [ + 159, + 330, + 175, + 341 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 330, + 505, + 342 + ], + "score": 1.0, + "content": ". 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SeedTest LossLoss (Key frequencies removed)|Loss (All other frequencies removed)
12.07:10-76.5:1005.7.10-8
22.1·10-71.1 · 1016.2.10-8
32.05·10-76.7·1005.5.10-8
42.33·10-76.8.1006.0:10-8
", + "type": "table", + "image_path": "194164cc08722526fb989c955cba31673937332873a42550216ea9e8c08d0820.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 125, + 364, + 485, + 382.6666666666667 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 125, + 382.6666666666667, + 485, + 401.33333333333337 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 125, + 401.33333333333337, + 485, + 420.00000000000006 + ], + "spans": [], + "index": 8 + } + ] + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 504, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 441 + ], + "score": 1.0, + "content": "Table 4: As discussed in Appendix C.2.1, ablating the key frequencies for each of the networks re-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 438, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 452 + ], + "score": 1.0, + "content": "duces performance to worse than chance, while ablating all other frequencies improves performance.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 472, + 505, + 539 + ], + "lines": [ + { + "bbox": [ + 106, + 471, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 485 + ], + "score": 1.0, + "content": "Confirming that the other seeds use the Fourier Multiplication Algorithm. In Figure 16, we", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 484, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 390, + 495 + ], + "score": 1.0, + "content": "show the norms of the Fourier components of the embedding matrix", + "type": "text" + }, + { + "bbox": [ + 390, + 484, + 407, + 495 + ], + "score": 0.89, + "content": "W _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 484, + 505, + 495 + ], + "score": 1.0, + "content": "for each of the 4 other", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "score": 1.0, + "content": "random seeds. As with the mainline model, the matrices are sparse in the Fourier basis. In Figure", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 506, + 504, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 398, + 517 + ], + "score": 1.0, + "content": "17, we show the norms of the Fourier components of the neuron-logit map", + "type": "text" + }, + { + "bbox": [ + 398, + 506, + 414, + 516 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 506, + 504, + 517 + ], + "score": 1.0, + "content": "for the 4 other random", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 517, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 505, + 528 + ], + "score": 1.0, + "content": "seeds. The matrices are sparse in the Fourier basis, enabling us to identify 3 or 4 key frequencies for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 528, + 401, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 401, + 539 + ], + "score": 1.0, + "content": "each of the seeds. Again, note that the specific frequencies differ by seed.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 544, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 543, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 557 + ], + "score": 1.0, + "content": "Using the key frequencies identified in the neuron-logit map, we repeat the experiment in Sec-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "tion 4.2, where we “read off” the MLP activations in the 6 or 8 directions corresponding to the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "key frequencies. As with our mainline model, this lets us identify the trigonometric identities for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 107, + 577, + 171, + 589 + ], + "score": 0.89, + "content": "\\cos { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 577, + 189, + 590 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 190, + 577, + 252, + 589 + ], + "score": 0.9, + "content": "\\sin { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "being computed at the MLP layer. We confirm that the trigono-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 588, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 600 + ], + "score": 1.0, + "content": "metric identities are a good approximation by approximating the activations with a single term of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 144, + 612 + ], + "score": 1.0, + "content": "the form", + "type": "text" + }, + { + "bbox": [ + 144, + 599, + 208, + 611 + ], + "score": 0.9, + "content": "\\cos { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 599, + 222, + 612 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 222, + 599, + 285, + 611 + ], + "score": 0.89, + "content": "\\sin \\left( w _ { k } ( a + b ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "—as with the mainline model, the fraction of variance", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 610, + 269, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 240, + 622 + ], + "score": 1.0, + "content": "explained is consistently close to", + "type": "text" + }, + { + "bbox": [ + 241, + 611, + 264, + 620 + ], + "score": 0.85, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 610, + 269, + 622 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 504, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "Next, we ablate the key frequencies from the logits as in Section 4.4 and report the results in Table", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "4. As with the mainline model, ablating all of the key frequencies reduces performance to worse", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 648, + 468, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 468, + 662 + ], + "score": 1.0, + "content": "than chance, while ablating everything but the key frequencies improves test performance.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "Progress measures and grokking. Finally, we confirm the progress measure and grokking results", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "from the mainline model on other runs with the same architecture. In Figure 18, we display the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "train, test, and restricted loss for each of the four other random seeds. In Figure 19, we display", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 402, + 711 + ], + "score": 1.0, + "content": "the Gini coefficients of the Fourier components of the embedding matrix", + "type": "text" + }, + { + "bbox": [ + 402, + 699, + 419, + 710 + ], + "score": 0.9, + "content": "W _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "and the neuron-logit", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 126, + 721 + ], + "score": 1.0, + "content": "map", + "type": "text" + }, + { + "bbox": [ + 126, + 710, + 143, + 721 + ], + "score": 0.9, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 709, + 505, + 721 + ], + "score": 1.0, + "content": "for each of the four other random seeds. The shape of the curves are very similar to those", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "of the mainline model, allowing us to divide grokking on these models into the same three phases", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + } + ], + "page_idx": 20, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 115, + 94, + 497, + 306 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 115, + 94, + 497, + 306 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 94, + 497, + 306 + ], + "spans": [ + { + "bbox": [ + 115, + 94, + 497, + 306 + ], + "score": 0.976, + "type": "image", + "image_path": "2049e7e629e850219ced239cd20167da1e3876d838ebe35a707f31334989eb9c.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 115, + 94, + 497, + 164.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 115, + 164.66666666666669, + 497, + 235.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 115, + 235.33333333333337, + 497, + 306.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 319, + 505, + 353 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "score": 1.0, + "content": "Figure 17: The norms of the direction corresponding to sine and cosine waves in the neuron-logit", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 158, + 342 + ], + "score": 1.0, + "content": "map weights", + "type": "text" + }, + { + "bbox": [ + 159, + 330, + 175, + 341 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 330, + 505, + 342 + ], + "score": 1.0, + "content": ". 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SeedTest LossLoss (Key frequencies removed)|Loss (All other frequencies removed)
12.07:10-76.5:1005.7.10-8
22.1·10-71.1 · 1016.2.10-8
32.05·10-76.7·1005.5.10-8
42.33·10-76.8.1006.0:10-8
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In Figure 16, we", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 484, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 390, + 495 + ], + "score": 1.0, + "content": "show the norms of the Fourier components of the embedding matrix", + "type": "text" + }, + { + "bbox": [ + 390, + 484, + 407, + 495 + ], + "score": 0.89, + "content": "W _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 484, + 505, + 495 + ], + "score": 1.0, + "content": "for each of the 4 other", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "score": 1.0, + "content": "random seeds. As with the mainline model, the matrices are sparse in the Fourier basis. In Figure", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 506, + 504, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 398, + 517 + ], + "score": 1.0, + "content": "17, we show the norms of the Fourier components of the neuron-logit map", + "type": "text" + }, + { + "bbox": [ + 398, + 506, + 414, + 516 + ], + "score": 0.89, + "content": "W _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 506, + 504, + 517 + ], + "score": 1.0, + "content": "for the 4 other random", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 517, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 505, + 528 + ], + "score": 1.0, + "content": "seeds. 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As with our mainline model, this lets us identify the trigonometric identities for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 107, + 577, + 171, + 589 + ], + "score": 0.89, + "content": "\\cos { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 577, + 189, + 590 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 190, + 577, + 252, + 589 + ], + "score": 0.9, + "content": "\\sin { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "being computed at the MLP layer. We confirm that the trigono-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 588, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 600 + ], + "score": 1.0, + "content": "metric identities are a good approximation by approximating the activations with a single term of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 144, + 612 + ], + "score": 1.0, + "content": "the form", + "type": "text" + }, + { + "bbox": [ + 144, + 599, + 208, + 611 + ], + "score": 0.9, + "content": "\\cos { ( w _ { k } ( a + b ) ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 599, + 222, + 612 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 222, + 599, + 285, + 611 + ], + "score": 0.89, + "content": "\\sin \\left( w _ { k } ( a + b ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "—as with the mainline model, the fraction of variance", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 610, + 269, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 240, + 622 + ], + "score": 1.0, + "content": "explained is consistently close to", + "type": "text" + }, + { + "bbox": [ + 241, + 611, + 264, + 620 + ], + "score": 0.85, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 610, + 269, + 622 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 543, + 506, + 622 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 504, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "Next, we ablate the key frequencies from the logits as in Section 4.4 and report the results in Table", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "4. As with the mainline model, ablating all of the key frequencies reduces performance to worse", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 648, + 468, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 468, + 662 + ], + "score": 1.0, + "content": "than chance, while ablating everything but the key frequencies improves test performance.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 627, + 505, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "Progress measures and grokking. 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The shape of the curves are very similar to those", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "of the mainline model, allowing us to divide grokking on these models into the same three phases", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 436, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 505, + 449 + ], + "score": 1.0, + "content": "identified in the main text. 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The lines delineate the 3 phases of training: memorization, circuit formation, and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "score": 1.0, + "content": "cleanup (and a final stable phase). As with the mainline model, restricted loss consistently declines", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 369, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 381 + ], + "score": 1.0, + "content": "prior to train loss. Note that while the shapes of the loss curves are similar to each other and those of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "the mainline model, the exact time that grokking occurs (and thus the dividers between the phases", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "of grokking) differ by random seed. Interestingly, memorization is complete by around 1400 steps", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 401, + 173, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 173, + 414 + ], + "score": 1.0, + "content": "for all five runs.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 108, + 437, + 504, + 459 + ], + "lines": [ + { + "bbox": [ + 106, + 436, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 505, + 449 + ], + "score": 1.0, + "content": "identified in the main text. 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We find that grokking occurs", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 251, + 554 + ], + "score": 1.0, + "content": "for the modular addition task with", + "type": "text" + }, + { + "bbox": [ + 251, + 542, + 293, + 552 + ], + "score": 0.89, + "content": "P = 1 1 3", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "for many data fractions (that is, the fraction of the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 147, + 564 + ], + "score": 0.69, + "content": "1 1 3 \\cdot 1 1 3", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "pairs of inputs that the model sees during training), as shown in Figure 20. Smaller", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 393, + 576 + ], + "score": 1.0, + "content": "amount lead to slower grokking, but sufficiently large fractions of data", + "type": "text" + }, + { + "bbox": [ + 394, + 564, + 427, + 575 + ], + "score": 0.85, + "content": "( \\geq 6 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 564, + 505, + 576 + ], + "score": 1.0, + "content": ") lead to immediate", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 575, + 294, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 294, + 587 + ], + "score": 1.0, + "content": "generalization, as shown in Figures 20 and 21.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 591, + 504, + 614 + ], + "lines": [ + { + "bbox": [ + 105, + 590, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 606 + ], + "score": 1.0, + "content": "As with the results in Appendix C.2.1, all of the 1-layer transformers in this section also converge", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 603, + 289, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 289, + 615 + ], + "score": 1.0, + "content": "to using the Fourier multiplication algorithm.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 628, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 106, + 628, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 505, + 640 + ], + "score": 1.0, + "content": "2-Layer Transformers. As shown in Figure 22, 2-layer transformers also exhibit some degree", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 640, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 651 + ], + "score": 1.0, + "content": "of grokking. However, this is complicated by the slingshot mechanism (Thilak et al., 2022). We", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "display the excluded loss of a 2-layer transformer in Figure 23 and find it shows a similar pattern to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 662, + 491, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 491, + 674 + ], + "score": 1.0, + "content": "the mainline 1-layer transformer, in that it improves relatively smoothly before grokking occurs.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 504, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 468, + 700 + ], + "score": 1.0, + "content": "Smaller and larger primes. We also examined smaller and larger prime moduli. For", + "type": "text" + }, + { + "bbox": [ + 468, + 687, + 504, + 698 + ], + "score": 0.87, + "content": "P = 5 3", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "(Figure 24), we explored a variety of weight decays to observe grokking in the small prime case.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 282, + 722 + ], + "score": 1.0, + "content": "With the original weight decay setting of", + "type": "text" + }, + { + "bbox": [ + 282, + 710, + 312, + 720 + ], + "score": 0.9, + "content": "\\lambda \\ : = \\ : 1", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 709, + 505, + 722 + ], + "score": 1.0, + "content": ", we found that the models never generalized.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 275, + 733 + ], + "score": 1.0, + "content": "However, increasing the weight decay to", + "type": "text" + }, + { + "bbox": [ + 275, + 721, + 303, + 731 + ], + "score": 0.88, + "content": "\\lambda = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "does allow the model to grok. 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The lines delineate the 3 phases of training: memorization, circuit formation, and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "score": 1.0, + "content": "cleanup (and a final stable phase). As with the mainline model, restricted loss consistently declines", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 369, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 381 + ], + "score": 1.0, + "content": "prior to train loss. Note that while the shapes of the loss curves are similar to each other and those of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "the mainline model, the exact time that grokking occurs (and thus the dividers between the phases", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "of grokking) differ by random seed. Interestingly, memorization is complete by around 1400 steps", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 401, + 173, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 173, + 414 + ], + "score": 1.0, + "content": "for all five runs.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 108, + 437, + 504, + 459 + ], + "lines": [], + "index": 10.5, + "bbox_fs": [ + 106, + 436, + 505, + 460 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 474, + 334, + 485 + ], + "lines": [ + { + "bbox": [ + 106, + 474, + 335, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 335, + 486 + ], + "score": 1.0, + "content": "C.2.2 RESULTS FOR OTHER EXPERIMENTAL SETUPS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 494, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "In this section, we provide further evidence that small transformers grok on the modular addition", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 505, + 495, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 482, + 517 + ], + "score": 1.0, + "content": "task, by varying the size of the network, the amount of training data, and the size of the prime", + "type": "text" + }, + { + "bbox": [ + 482, + 505, + 491, + 515 + ], + "score": 0.8, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 505, + 495, + 517 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 493, + 505, + 517 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 531, + 505, + 586 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "1-Layer Transformers with Varying Fractions of Training Data. We find that grokking occurs", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 251, + 554 + ], + "score": 1.0, + "content": "for the modular addition task with", + "type": "text" + }, + { + "bbox": [ + 251, + 542, + 293, + 552 + ], + "score": 0.89, + "content": "P = 1 1 3", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "for many data fractions (that is, the fraction of the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 147, + 564 + ], + "score": 0.69, + "content": "1 1 3 \\cdot 1 1 3", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "pairs of inputs that the model sees during training), as shown in Figure 20. Smaller", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 393, + 576 + ], + "score": 1.0, + "content": "amount lead to slower grokking, but sufficiently large fractions of data", + "type": "text" + }, + { + "bbox": [ + 394, + 564, + 427, + 575 + ], + "score": 0.85, + "content": "( \\geq 6 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 564, + 505, + 576 + ], + "score": 1.0, + "content": ") lead to immediate", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 575, + 294, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 294, + 587 + ], + "score": 1.0, + "content": "generalization, as shown in Figures 20 and 21.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 531, + 505, + 587 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 591, + 504, + 614 + ], + "lines": [ + { + "bbox": [ + 105, + 590, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 606 + ], + "score": 1.0, + "content": "As with the results in Appendix C.2.1, all of the 1-layer transformers in this section also converge", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 603, + 289, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 289, + 615 + ], + "score": 1.0, + "content": "to using the Fourier multiplication algorithm.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 590, + 505, + 615 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 628, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 106, + 628, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 505, + 640 + ], + "score": 1.0, + "content": "2-Layer Transformers. As shown in Figure 22, 2-layer transformers also exhibit some degree", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 640, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 651 + ], + "score": 1.0, + "content": "of grokking. However, this is complicated by the slingshot mechanism (Thilak et al., 2022). We", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "display the excluded loss of a 2-layer transformer in Figure 23 and find it shows a similar pattern to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 662, + 491, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 491, + 674 + ], + "score": 1.0, + "content": "the mainline 1-layer transformer, in that it improves relatively smoothly before grokking occurs.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 628, + 505, + 674 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 504, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 468, + 700 + ], + "score": 1.0, + "content": "Smaller and larger primes. We also examined smaller and larger prime moduli. For", + "type": "text" + }, + { + "bbox": [ + 468, + 687, + 504, + 698 + ], + "score": 0.87, + "content": "P = 5 3", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "(Figure 24), we explored a variety of weight decays to observe grokking in the small prime case.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 282, + 722 + ], + "score": 1.0, + "content": "With the original weight decay setting of", + "type": "text" + }, + { + "bbox": [ + 282, + 710, + 312, + 720 + ], + "score": 0.9, + "content": "\\lambda \\ : = \\ : 1", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 709, + 505, + 722 + ], + "score": 1.0, + "content": ", we found that the models never generalized.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 275, + 733 + ], + "score": 1.0, + "content": "However, increasing the weight decay to", + "type": "text" + }, + { + "bbox": [ + 275, + 721, + 303, + 731 + ], + "score": 0.88, + "content": "\\lambda = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "does allow the model to grok. 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We", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "believe this is because the amount of data seen by the model is greatly increased compared to the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 159, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 107, + 160, + 146, + 171 + ], + "score": 0.89, + "content": "P = 1 1 3", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 159, + 192, + 173 + ], + "score": 1.0, + "content": "case (from", + "type": "text" + }, + { + "bbox": [ + 193, + 160, + 213, + 171 + ], + "score": 0.86, + "content": "30 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 159, + 225, + 173 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 226, + 160, + 265, + 171 + ], + "score": 0.76, + "content": "1 1 3 \\cdot 1 1 3", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 159, + 299, + 173 + ], + "score": 1.0, + "content": "pairs to", + "type": "text" + }, + { + "bbox": [ + 299, + 160, + 319, + 171 + ], + "score": 0.87, + "content": "30 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 159, + 331, + 173 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 332, + 160, + 371, + 171 + ], + "score": 0.79, + "content": "4 0 1 \\cdot 4 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 159, + 505, + 173 + ], + "score": 1.0, + "content": "pairs), thereby favoring the gen-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 171, + 504, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 392, + 182 + ], + "score": 1.0, + "content": "eralizing solution from the start. We then trained 3 models each using", + "type": "text" + }, + { + "bbox": [ + 393, + 171, + 407, + 182 + ], + "score": 0.85, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 172, + 411, + 182 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 412, + 171, + 431, + 182 + ], + "score": 0.85, + "content": "1 0 \\dot { \\% }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 172, + 435, + 182 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 435, + 171, + 455, + 182 + ], + "score": 0.87, + "content": "2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 172, + 504, + 182 + ], + "score": 1.0, + "content": "of the pairs", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 504, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 192, + 194 + ], + "score": 1.0, + "content": "of training data with", + "type": "text" + }, + { + "bbox": [ + 192, + 182, + 219, + 192 + ], + "score": 0.9, + "content": "\\lambda = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 182, + 375, + 194 + ], + "score": 1.0, + "content": ", and found that the models trained on", + "type": "text" + }, + { + "bbox": [ + 375, + 182, + 390, + 193 + ], + "score": 0.88, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 182, + 409, + 194 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 409, + 182, + 429, + 193 + ], + "score": 0.88, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 182, + 504, + 194 + ], + "score": 1.0, + "content": "of the data imme-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 193, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 371, + 205 + ], + "score": 1.0, + "content": "diately overfit and never generalized, while the models trained on", + "type": "text" + }, + { + "bbox": [ + 372, + 193, + 391, + 204 + ], + "score": 0.87, + "content": "2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 193, + 505, + 205 + ], + "score": 1.0, + "content": "of the data also generalized", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 203, + 161, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 161, + 217 + ], + "score": 1.0, + "content": "immediately.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 233, + 470, + 255 + ], + "lines": [ + { + "bbox": [ + 105, + 231, + 471, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 471, + 247 + ], + "score": 1.0, + "content": "C.2.3 GENERALIZING MODELS CONSISTENTLY USE THE FOURIER MULTIPLICATION", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 140, + 244, + 197, + 256 + ], + "spans": [ + { + "bbox": [ + 140, + 244, + 197, + 256 + ], + "score": 1.0, + "content": "ALGORITHM", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 265, + 504, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 264, + 504, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 504, + 277 + ], + "score": 1.0, + "content": "For each of the models in Appendix C.2.2 that achieve low test loss, we repeated the analysis per-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 277, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 289 + ], + "score": 1.0, + "content": "formed in the mainline model, and summarize the results in Table 5. We list their key frequencies,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "Gini coefficients, and relevant FVEs. We find that every model trained with weight decay and that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 299, + 461, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 461, + 311 + ], + "score": 1.0, + "content": "generalizes correctly implements some variation of the Fourier multiplication algorithm.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 315, + 505, + 381 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 328 + ], + "score": 1.0, + "content": "Interestingly, the embedding and unembedding matrices of the models trained with dropout are not", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 326, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 300, + 339 + ], + "score": 1.0, + "content": "sparse in the Fourier basis, and the logits for the", + "type": "text" + }, + { + "bbox": [ + 300, + 327, + 333, + 338 + ], + "score": 0.9, + "content": "p = 0 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 326, + 506, + 339 + ], + "score": 1.0, + "content": "models are not as well explained by a sum", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 337, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 305, + 350 + ], + "score": 1.0, + "content": "of cosines as the other models (likely because the", + "type": "text" + }, + { + "bbox": [ + 305, + 338, + 338, + 349 + ], + "score": 0.9, + "content": "p = 0 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 337, + 505, + 350 + ], + "score": 1.0, + "content": "models are simply worse at the task). We", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "score": 1.0, + "content": "speculate that this is likely due to a combination of insufficient training epochs (as dropout models", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 360, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 371 + ], + "score": 1.0, + "content": "seem to take much longer to grok) and the inherent need for redundancy for networks trained via", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 370, + 143, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 143, + 383 + ], + "score": 1.0, + "content": "dropout.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 387, + 504, + 420 + ], + "lines": [ + { + "bbox": [ + 105, + 386, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 506, + 400 + ], + "score": 1.0, + "content": "As with the mainline model, we ignore the final skip connection (around the final MLP), as all of the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "score": 1.0, + "content": "generalizing models studied do not suffer significant performance penalties if the skip connection is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 408, + 234, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 234, + 422 + ], + "score": 1.0, + "content": "zero or mean ablated (Table 6).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 107, + 443, + 320, + 454 + ], + "lines": [ + { + "bbox": [ + 105, + 441, + 321, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 321, + 457 + ], + "score": 1.0, + "content": "D ADDITIONAL RESULTS ON GROKKING", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 470, + 462, + 481 + ], + "lines": [ + { + "bbox": [ + 106, + 470, + 464, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 464, + 482 + ], + "score": 1.0, + "content": "D.1 BOTH REGULARIZATION AND LIMITED DATA ARE NECESSARY FOR GROKKING", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 492, + 505, + 559 + ], + "lines": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "As discussed in Section 7 and Appendix C.2, the weight decay and the amount of data seem to have", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 505, + 504, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 504, + 516 + ], + "score": 1.0, + "content": "a strong effect on whether grokking occurs. To confirm this, we experiment with removing weight", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 513, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 506, + 529 + ], + "score": 1.0, + "content": "decay and varying the amount of data on 1-layer transformers. In Figure 26, we give the training,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 525, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 340, + 539 + ], + "score": 1.0, + "content": "test, and full excluded loss for a typical training run with", + "type": "text" + }, + { + "bbox": [ + 341, + 526, + 368, + 536 + ], + "score": 0.9, + "content": "\\lambda = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 525, + 505, + 539 + ], + "score": 1.0, + "content": "(no weight decay). As the figure", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "shows, no grokking occurs, and excluded loss does not increase, suggesting that the model does not", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 547, + 303, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 303, + 560 + ], + "score": 1.0, + "content": "form the circuit for generalizing algorithm at all.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 564, + 504, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 443, + 577 + ], + "score": 1.0, + "content": "In Figure 20, we show the test loss curves for models trained with weight decay", + "type": "text" + }, + { + "bbox": [ + 443, + 565, + 473, + 575 + ], + "score": 0.9, + "content": "\\lambda = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "and on", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 574, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 589 + ], + "score": 1.0, + "content": "various fractions of the data. Though all the train losses are approximately the same—that is, they", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 587, + 469, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 469, + 599 + ], + "score": 1.0, + "content": "memorize at the same rate, models trained on smaller fractions of data take longer to grok.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 603, + 505, + 659 + ], + "lines": [ + { + "bbox": [ + 106, + 604, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 383, + 615 + ], + "score": 1.0, + "content": "In Figure 27, we display the test and train loss of models trained with", + "type": "text" + }, + { + "bbox": [ + 383, + 604, + 416, + 614 + ], + "score": 0.9, + "content": "\\lambda = 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 604, + 434, + 615 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 434, + 604, + 467, + 614 + ], + "score": 0.89, + "content": "\\lambda = 3 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 604, + 505, + 615 + ], + "score": 1.0, + "content": ". Smaller", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 614, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 506, + 627 + ], + "score": 1.0, + "content": "amounts of weight decay lead to slower grokking, while larger amounts of weight decay lead to faster", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 625, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 465, + 639 + ], + "score": 1.0, + "content": "grokking—on average, it takes around 3k epochs for models to grok with weight decay", + "type": "text" + }, + { + "bbox": [ + 466, + 626, + 501, + 636 + ], + "score": 0.88, + "content": "\\lambda = 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 625, + 505, + 639 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 636, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 132, + 647 + ], + "score": 0.29, + "content": "5 { - } 1 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 636, + 331, + 649 + ], + "score": 1.0, + "content": "epochs for the models to grok with weight decay", + "type": "text" + }, + { + "bbox": [ + 331, + 637, + 365, + 647 + ], + "score": 0.9, + "content": "\\lambda = 1 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 636, + 386, + 649 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 386, + 637, + 403, + 647 + ], + "score": 0.36, + "content": "2 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 636, + 505, + 649 + ], + "score": 1.0, + "content": "epochs for the models to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 647, + 239, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 203, + 660 + ], + "score": 1.0, + "content": "grok with weight decay", + "type": "text" + }, + { + "bbox": [ + 203, + 648, + 236, + 658 + ], + "score": 0.89, + "content": "\\lambda = 3 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 647, + 239, + 660 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 664, + 504, + 687 + ], + "lines": [ + { + "bbox": [ + 106, + 664, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 505, + 677 + ], + "score": 1.0, + "content": "Finally, we test whether other forms of regularization can also induce grokking. We replaced weight", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 676, + 496, + 688 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 496, + 688 + ], + "score": 1.0, + "content": "decay with the following types of regularization while keeping all other hyperpameters the same:", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5 + }, + { + "type": "text", + "bbox": [ + 131, + 699, + 504, + 734 + ], + "lines": [ + { + "bbox": [ + 129, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 129, + 698, + 482, + 712 + ], + "score": 1.0, + "content": "1. Dropout We add dropout Srivastava et al. (2014) to the MLP neurons, with", + "type": "text" + }, + { + "bbox": [ + 482, + 699, + 505, + 711 + ], + "score": 0.85, + "content": "p \\in", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 143, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 143, + 709, + 200, + 722 + ], + "score": 0.85, + "content": "\\{ 0 . 2 , 0 . 5 , 0 . 8 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 708, + 469, + 723 + ], + "score": 1.0, + "content": ". That is, for each individual neuron, we set it to 0 with probability", + "type": "text" + }, + { + "bbox": [ + 469, + 711, + 475, + 721 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "during", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 140, + 715, + 415, + 738 + ], + "spans": [ + { + "bbox": [ + 140, + 715, + 392, + 738 + ], + "score": 1.0, + "content": "training, and also multiply the outputs of the other neurons by", + "type": "text" + }, + { + "bbox": [ + 392, + 720, + 409, + 735 + ], + "score": 0.92, + "content": "\\frac { 1 } { 1 - p }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 715, + 415, + 738 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45 + } + ], + "page_idx": 25, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 752, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 505, + 106 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 108, + 110, + 413, + 122 + ], + "lines": [ + { + "bbox": [ + 105, + 109, + 415, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 122, + 123 + ], + "score": 1.0, + "content": "For", + "type": "text" + }, + { + "bbox": [ + 123, + 111, + 160, + 121 + ], + "score": 0.89, + "content": "P = 1 0 9", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 109, + 415, + 123 + ], + "score": 1.0, + "content": ", we saw exactly the same behavior as with the mainline model.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 109, + 415, + 123 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 127, + 505, + 216 + ], + "lines": [ + { + "bbox": [ + 105, + 126, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 123, + 141 + ], + "score": 1.0, + "content": "For", + "type": "text" + }, + { + "bbox": [ + 123, + 128, + 163, + 138 + ], + "score": 0.88, + "content": "P = 4 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 126, + 505, + 141 + ], + "score": 1.0, + "content": "(Figure 25), we could not get grokking, even by varying the weight decay parame-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 120, + 150 + ], + "score": 1.0, + "content": "ter", + "type": "text" + }, + { + "bbox": [ + 120, + 138, + 219, + 150 + ], + "score": 0.91, + "content": "\\lambda \\in \\{ 0 . 3 , 0 . 5 , 1 , 3 , 5 , 8 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 138, + 505, + 150 + ], + "score": 1.0, + "content": ". Instead, the model immediately learns the generalizing solution. We", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "believe this is because the amount of data seen by the model is greatly increased compared to the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 159, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 107, + 160, + 146, + 171 + ], + "score": 0.89, + "content": "P = 1 1 3", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 159, + 192, + 173 + ], + "score": 1.0, + "content": "case (from", + "type": "text" + }, + { + "bbox": [ + 193, + 160, + 213, + 171 + ], + "score": 0.86, + "content": "30 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 159, + 225, + 173 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 226, + 160, + 265, + 171 + ], + "score": 0.76, + "content": "1 1 3 \\cdot 1 1 3", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 159, + 299, + 173 + ], + "score": 1.0, + "content": "pairs to", + "type": "text" + }, + { + "bbox": [ + 299, + 160, + 319, + 171 + ], + "score": 0.87, + "content": "30 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 159, + 331, + 173 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 332, + 160, + 371, + 171 + ], + "score": 0.79, + "content": "4 0 1 \\cdot 4 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 159, + 505, + 173 + ], + "score": 1.0, + "content": "pairs), thereby favoring the gen-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 171, + 504, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 392, + 182 + ], + "score": 1.0, + "content": "eralizing solution from the start. We then trained 3 models each using", + "type": "text" + }, + { + "bbox": [ + 393, + 171, + 407, + 182 + ], + "score": 0.85, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 172, + 411, + 182 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 412, + 171, + 431, + 182 + ], + "score": 0.85, + "content": "1 0 \\dot { \\% }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 172, + 435, + 182 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 435, + 171, + 455, + 182 + ], + "score": 0.87, + "content": "2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 172, + 504, + 182 + ], + "score": 1.0, + "content": "of the pairs", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 504, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 192, + 194 + ], + "score": 1.0, + "content": "of training data with", + "type": "text" + }, + { + "bbox": [ + 192, + 182, + 219, + 192 + ], + "score": 0.9, + "content": "\\lambda = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 182, + 375, + 194 + ], + "score": 1.0, + "content": ", and found that the models trained on", + "type": "text" + }, + { + "bbox": [ + 375, + 182, + 390, + 193 + ], + "score": 0.88, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 182, + 409, + 194 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 409, + 182, + 429, + 193 + ], + "score": 0.88, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 182, + 504, + 194 + ], + "score": 1.0, + "content": "of the data imme-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 193, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 371, + 205 + ], + "score": 1.0, + "content": "diately overfit and never generalized, while the models trained on", + "type": "text" + }, + { + "bbox": [ + 372, + 193, + 391, + 204 + ], + "score": 0.87, + "content": "2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 193, + 505, + 205 + ], + "score": 1.0, + "content": "of the data also generalized", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 203, + 161, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 161, + 217 + ], + "score": 1.0, + "content": "immediately.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 126, + 505, + 217 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 233, + 470, + 255 + ], + "lines": [ + { + "bbox": [ + 105, + 231, + 471, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 471, + 247 + ], + "score": 1.0, + "content": "C.2.3 GENERALIZING MODELS CONSISTENTLY USE THE FOURIER MULTIPLICATION", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 140, + 244, + 197, + 256 + ], + "spans": [ + { + "bbox": [ + 140, + 244, + 197, + 256 + ], + "score": 1.0, + "content": "ALGORITHM", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 231, + 471, + 256 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 265, + 504, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 264, + 504, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 504, + 277 + ], + "score": 1.0, + "content": "For each of the models in Appendix C.2.2 that achieve low test loss, we repeated the analysis per-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 277, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 289 + ], + "score": 1.0, + "content": "formed in the mainline model, and summarize the results in Table 5. We list their key frequencies,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "Gini coefficients, and relevant FVEs. We find that every model trained with weight decay and that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 299, + 461, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 461, + 311 + ], + "score": 1.0, + "content": "generalizes correctly implements some variation of the Fourier multiplication algorithm.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 264, + 505, + 311 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 315, + 505, + 381 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 328 + ], + "score": 1.0, + "content": "Interestingly, the embedding and unembedding matrices of the models trained with dropout are not", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 326, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 300, + 339 + ], + "score": 1.0, + "content": "sparse in the Fourier basis, and the logits for the", + "type": "text" + }, + { + "bbox": [ + 300, + 327, + 333, + 338 + ], + "score": 0.9, + "content": "p = 0 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 326, + 506, + 339 + ], + "score": 1.0, + "content": "models are not as well explained by a sum", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 337, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 305, + 350 + ], + "score": 1.0, + "content": "of cosines as the other models (likely because the", + "type": "text" + }, + { + "bbox": [ + 305, + 338, + 338, + 349 + ], + "score": 0.9, + "content": "p = 0 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 337, + 505, + 350 + ], + "score": 1.0, + "content": "models are simply worse at the task). We", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "score": 1.0, + "content": "speculate that this is likely due to a combination of insufficient training epochs (as dropout models", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 360, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 371 + ], + "score": 1.0, + "content": "seem to take much longer to grok) and the inherent need for redundancy for networks trained via", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 370, + 143, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 143, + 383 + ], + "score": 1.0, + "content": "dropout.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 315, + 506, + 383 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 387, + 504, + 420 + ], + "lines": [ + { + "bbox": [ + 105, + 386, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 506, + 400 + ], + "score": 1.0, + "content": "As with the mainline model, we ignore the final skip connection (around the final MLP), as all of the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "score": 1.0, + "content": "generalizing models studied do not suffer significant performance penalties if the skip connection is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 408, + 234, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 234, + 422 + ], + "score": 1.0, + "content": "zero or mean ablated (Table 6).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 386, + 506, + 422 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 443, + 320, + 454 + ], + "lines": [ + { + "bbox": [ + 105, + 441, + 321, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 321, + 457 + ], + "score": 1.0, + "content": "D ADDITIONAL RESULTS ON GROKKING", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 470, + 462, + 481 + ], + "lines": [ + { + "bbox": [ + 106, + 470, + 464, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 464, + 482 + ], + "score": 1.0, + "content": "D.1 BOTH REGULARIZATION AND LIMITED DATA ARE NECESSARY FOR GROKKING", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 106, + 470, + 464, + 482 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 492, + 505, + 559 + ], + "lines": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "As discussed in Section 7 and Appendix C.2, the weight decay and the amount of data seem to have", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 505, + 504, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 504, + 516 + ], + "score": 1.0, + "content": "a strong effect on whether grokking occurs. To confirm this, we experiment with removing weight", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 513, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 506, + 529 + ], + "score": 1.0, + "content": "decay and varying the amount of data on 1-layer transformers. In Figure 26, we give the training,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 525, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 340, + 539 + ], + "score": 1.0, + "content": "test, and full excluded loss for a typical training run with", + "type": "text" + }, + { + "bbox": [ + 341, + 526, + 368, + 536 + ], + "score": 0.9, + "content": "\\lambda = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 525, + 505, + 539 + ], + "score": 1.0, + "content": "(no weight decay). As the figure", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "shows, no grokking occurs, and excluded loss does not increase, suggesting that the model does not", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 547, + 303, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 303, + 560 + ], + "score": 1.0, + "content": "form the circuit for generalizing algorithm at all.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 493, + 506, + 560 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 564, + 504, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 443, + 577 + ], + "score": 1.0, + "content": "In Figure 20, we show the test loss curves for models trained with weight decay", + "type": "text" + }, + { + "bbox": [ + 443, + 565, + 473, + 575 + ], + "score": 0.9, + "content": "\\lambda = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "and on", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 574, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 589 + ], + "score": 1.0, + "content": "various fractions of the data. Though all the train losses are approximately the same—that is, they", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 587, + 469, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 469, + 599 + ], + "score": 1.0, + "content": "memorize at the same rate, models trained on smaller fractions of data take longer to grok.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 564, + 506, + 599 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 603, + 505, + 659 + ], + "lines": [ + { + "bbox": [ + 106, + 604, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 383, + 615 + ], + "score": 1.0, + "content": "In Figure 27, we display the test and train loss of models trained with", + "type": "text" + }, + { + "bbox": [ + 383, + 604, + 416, + 614 + ], + "score": 0.9, + "content": "\\lambda = 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 604, + 434, + 615 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 434, + 604, + 467, + 614 + ], + "score": 0.89, + "content": "\\lambda = 3 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 604, + 505, + 615 + ], + "score": 1.0, + "content": ". Smaller", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 614, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 506, + 627 + ], + "score": 1.0, + "content": "amounts of weight decay lead to slower grokking, while larger amounts of weight decay lead to faster", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 625, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 465, + 639 + ], + "score": 1.0, + "content": "grokking—on average, it takes around 3k epochs for models to grok with weight decay", + "type": "text" + }, + { + "bbox": [ + 466, + 626, + 501, + 636 + ], + "score": 0.88, + "content": "\\lambda = 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 625, + 505, + 639 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 636, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 132, + 647 + ], + "score": 0.29, + "content": "5 { - } 1 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 636, + 331, + 649 + ], + "score": 1.0, + "content": "epochs for the models to grok with weight decay", + "type": "text" + }, + { + "bbox": [ + 331, + 637, + 365, + 647 + ], + "score": 0.9, + "content": "\\lambda = 1 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 636, + 386, + 649 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 386, + 637, + 403, + 647 + ], + "score": 0.36, + "content": "2 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 636, + 505, + 649 + ], + "score": 1.0, + "content": "epochs for the models to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 647, + 239, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 203, + 660 + ], + "score": 1.0, + "content": "grok with weight decay", + "type": "text" + }, + { + "bbox": [ + 203, + 648, + 236, + 658 + ], + "score": 0.89, + "content": "\\lambda = 3 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 647, + 239, + 660 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 604, + 506, + 660 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 664, + 504, + 687 + ], + "lines": [ + { + "bbox": [ + 106, + 664, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 505, + 677 + ], + "score": 1.0, + "content": "Finally, we test whether other forms of regularization can also induce grokking. We replaced weight", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 676, + 496, + 688 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 496, + 688 + ], + "score": 1.0, + "content": "decay with the following types of regularization while keeping all other hyperpameters the same:", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5, + "bbox_fs": [ + 106, + 664, + 505, + 688 + ] + }, + { + "type": "text", + "bbox": [ + 131, + 699, + 504, + 734 + ], + "lines": [ + { + "bbox": [ + 129, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 129, + 698, + 482, + 712 + ], + "score": 1.0, + "content": "1. Dropout We add dropout Srivastava et al. (2014) to the MLP neurons, with", + "type": "text" + }, + { + "bbox": [ + 482, + 699, + 505, + 711 + ], + "score": 0.85, + "content": "p \\in", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 143, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 143, + 709, + 200, + 722 + ], + "score": 0.85, + "content": "\\{ 0 . 2 , 0 . 5 , 0 . 8 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 708, + 469, + 723 + ], + "score": 1.0, + "content": ". 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ModelTest LossGini(Wε)Gini(WL)Key FrequenciesLogit FVEMLP FVE
40% Training Data1.98:10-70.760.79[17,43,49,55]94.9%83.3% [26.1%]
50% Training Data1.68 : 10-70.750.77[2,17,31,41, 44]91.2%85.2% [28.2%]
60% Training Data1.23.10-70.790.84[2,23,34,51]96.4%95.7% [1.4%]
70% Training Data9.85.10-80.800.91[14,15,26]99.0%98.9% [0.4%]
80% Training Data5.83.10-70.620.80[38,41]63.9%94.1% [2.5%]
90% Training Data1.11:10-70.790.88[3,26,34,43]98.6%98.7% [0.3%]
2Layer Transformer9.54:10-70.590.80[14,18,29]91.8%95.2%[1.9%]
2 Layer Transformer4.41:10-50.550.73[7,12,35,49]86.1%86.2% [6.4%]
2 Layer Transformer6.50.10-20.660.80[4,9,28]88.5%85.4% [5.9%]
2 Layer Transformer4.18.10-20.560.76[4,5,15,54]91.4%81.2% [17.8%]
2 Layer Transformer1.75:10-20.680.71[3,4,13,30,38]84.0%71.9% [19.5%]
P=533.00:10-40.610.68[6,9,16,21]91.2%90.2% [5.8%]
P=531.03:10-40.560.72[4,13,16]94.8%93.1% [6.4%]
P=531.21:10-50.660.79[13,22,23]98.2%97.6% [0.9%]
P=533.95·10-60.660.74[3,14,15]88.5%91.8% [4.6%]
P=535.56:10-60.670.80[10,14,22]98.1%98.3% [0.6%]
P=1092.02:10-70.760.83[6,7,22,25]98.0%97.3% [1.9%]
P=1092.95.10-70.690.82[8,14,29,32,41]95.2%94.7% [2.3%]
P=1091.66·10-70.780.86[13,23,39,45]98.5%97.6% [0.9%]
P=1092.50.10-70.680.82[8,13,32,41]96.8%95.5% [2.3%]
P=1092.77:10-70.760.85[29,37,38,49]97.9%98.1% [0.8%]
Dropout p= 0.22.65:10-10.190.46[1,4,7,17,22,33,40,49,55]71.3%65.0% [17.5%]
Dropout p = 0.24.52.10-10.190.46[3,8,19,28,32,34,40,44]73.3%71.4% [10.7%]
Dropout p = 0.22.03:10-10.200.45[4,5,32,38,41,44,49,50]74.2%71.1% [10.6%]
Dropout p = 0.5<10-80.260.56[1,4,26,46,47,55]89.4%88.9%[3.5%]
Dropout p = 0.52.01.10-20.200.49[16,21,35,47,53]88.4%88.4% [3.0%]
Dropout p = 0.5<10-80.250.54[1,4,7,19,29,31,42]86.1%85.6%[4.0%]
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Numbers in square brackets represent the standard deviation. For 2 Layer models, we", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 458, + 308, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 262, + 471 + ], + "score": 1.0, + "content": "use the final layer MLP activations for", + "type": "text" + }, + { + "bbox": [ + 262, + 459, + 305, + 471 + ], + "score": 0.9, + "content": "\\textstyle \\mathrm { M L P } ( a , b )", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 458, + 308, + 471 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 480, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 106, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 439, + 493 + ], + "score": 1.0, + "content": "We omit test accuracy because every model on this list except for the dropout", + "type": "text" + }, + { + "bbox": [ + 440, + 481, + 479, + 492 + ], + "score": 0.88, + "content": "p \\ = \\ 0 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "mod-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 491, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 156, + 504 + ], + "score": 1.0, + "content": "els achieves", + "type": "text" + }, + { + "bbox": [ + 157, + 491, + 200, + 502 + ], + "score": 0.9, + "content": "> 9 9 . 9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 491, + 331, + 504 + ], + "score": 1.0, + "content": "test accuracy, while the dropout", + "type": "text" + }, + { + "bbox": [ + 331, + 492, + 364, + 503 + ], + "score": 0.9, + "content": "p = 0 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 491, + 460, + 504 + ], + "score": 1.0, + "content": "models achieve around", + "type": "text" + }, + { + "bbox": [ + 460, + 492, + 487, + 502 + ], + "score": 0.87, + "content": "9 9 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 491, + 506, + 504 + ], + "score": 1.0, + "content": "test", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 502, + 147, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 147, + 516 + ], + "score": 1.0, + "content": "accuracy.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "table", + "bbox": [ + 107, + 566, + 533, + 644 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 107, + 566, + 533, + 644 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 107, + 566, + 533, + 644 + ], + "spans": [ + { + "bbox": [ + 107, + 566, + 533, + 644 + ], + "score": 0.98, + "html": "
Model TypeLossAccuracyAblated LossAblated Acuracy
VaryingData Fraction1.83:10-7 (1.65:10 7100%7.74· 10-7 (6.74· 10-7)100%
2 Layer Transformer1.97·10-2 (2.41·10-299.6%4.63· 10-2 (6.72 : 10-298.7%
P=535.96·10-5 (8.91·10-5)100%1.5·10-4 (2.70· 10-4)100%
P=1091.94· 10-7 (3.74· 10-8)100%6.53· 10-7 (1.41 · 10-7)100%
Dropout p = 0.20.215 (0.091)99.7%0.205 (0.075)99.7%
Dropout p = 0.54.68 · 10-³ (8.11· 10-3)100%3.6:10-3 (5.82.10-3)100%
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ModelTest LossGini(Wε)Gini(WL)Key FrequenciesLogit FVEMLP FVE
40% Training Data1.98:10-70.760.79[17,43,49,55]94.9%83.3% [26.1%]
50% Training Data1.68 : 10-70.750.77[2,17,31,41, 44]91.2%85.2% [28.2%]
60% Training Data1.23.10-70.790.84[2,23,34,51]96.4%95.7% [1.4%]
70% Training Data9.85.10-80.800.91[14,15,26]99.0%98.9% [0.4%]
80% Training Data5.83.10-70.620.80[38,41]63.9%94.1% [2.5%]
90% Training Data1.11:10-70.790.88[3,26,34,43]98.6%98.7% [0.3%]
2Layer Transformer9.54:10-70.590.80[14,18,29]91.8%95.2%[1.9%]
2 Layer Transformer4.41:10-50.550.73[7,12,35,49]86.1%86.2% [6.4%]
2 Layer Transformer6.50.10-20.660.80[4,9,28]88.5%85.4% [5.9%]
2 Layer Transformer4.18.10-20.560.76[4,5,15,54]91.4%81.2% [17.8%]
2 Layer Transformer1.75:10-20.680.71[3,4,13,30,38]84.0%71.9% [19.5%]
P=533.00:10-40.610.68[6,9,16,21]91.2%90.2% [5.8%]
P=531.03:10-40.560.72[4,13,16]94.8%93.1% [6.4%]
P=531.21:10-50.660.79[13,22,23]98.2%97.6% [0.9%]
P=533.95·10-60.660.74[3,14,15]88.5%91.8% [4.6%]
P=535.56:10-60.670.80[10,14,22]98.1%98.3% [0.6%]
P=1092.02:10-70.760.83[6,7,22,25]98.0%97.3% [1.9%]
P=1092.95.10-70.690.82[8,14,29,32,41]95.2%94.7% [2.3%]
P=1091.66·10-70.780.86[13,23,39,45]98.5%97.6% [0.9%]
P=1092.50.10-70.680.82[8,13,32,41]96.8%95.5% [2.3%]
P=1092.77:10-70.760.85[29,37,38,49]97.9%98.1% [0.8%]
Dropout p= 0.22.65:10-10.190.46[1,4,7,17,22,33,40,49,55]71.3%65.0% [17.5%]
Dropout p = 0.24.52.10-10.190.46[3,8,19,28,32,34,40,44]73.3%71.4% [10.7%]
Dropout p = 0.22.03:10-10.200.45[4,5,32,38,41,44,49,50]74.2%71.1% [10.6%]
Dropout p = 0.5<10-80.260.56[1,4,26,46,47,55]89.4%88.9%[3.5%]
Dropout p = 0.52.01.10-20.200.49[16,21,35,47,53]88.4%88.4% [3.0%]
Dropout p = 0.5<10-80.250.54[1,4,7,19,29,31,42]86.1%85.6%[4.0%]
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Numbers in square brackets represent the standard deviation. For 2 Layer models, we", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 458, + 308, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 262, + 471 + ], + "score": 1.0, + "content": "use the final layer MLP activations for", + "type": "text" + }, + { + "bbox": [ + 262, + 459, + 305, + 471 + ], + "score": 0.9, + "content": "\\textstyle \\mathrm { M L P } ( a , b )", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 458, + 308, + 471 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 413, + 506, + 471 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 480, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 106, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 439, + 493 + ], + "score": 1.0, + "content": "We omit test accuracy because every model on this list except for the dropout", + "type": "text" + }, + { + "bbox": [ + 440, + 481, + 479, + 492 + ], + "score": 0.88, + "content": "p \\ = \\ 0 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "mod-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 491, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 156, + 504 + ], + "score": 1.0, + "content": "els achieves", + "type": "text" + }, + { + "bbox": [ + 157, + 491, + 200, + 502 + ], + "score": 0.9, + "content": "> 9 9 . 9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 491, + 331, + 504 + ], + "score": 1.0, + "content": "test accuracy, while the dropout", + "type": "text" + }, + { + "bbox": [ + 331, + 492, + 364, + 503 + ], + "score": 0.9, + "content": "p = 0 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 491, + 460, + 504 + ], + "score": 1.0, + "content": "models achieve around", + "type": "text" + }, + { + "bbox": [ + 460, + 492, + 487, + 502 + ], + "score": 0.87, + "content": "9 9 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 491, + 506, + 504 + ], + "score": 1.0, + "content": "test", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 502, + 147, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 147, + 516 + ], + "score": 1.0, + "content": "accuracy.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 480, + 506, + 516 + ] + }, + { + "type": "table", + "bbox": [ + 107, + 566, + 533, + 644 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 107, + 566, + 533, + 644 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 107, + 566, + 533, + 644 + ], + "spans": [ + { + "bbox": [ + 107, + 566, + 533, + 644 + ], + "score": 0.98, + "html": "
Model TypeLossAccuracyAblated LossAblated Acuracy
VaryingData Fraction1.83:10-7 (1.65:10 7100%7.74· 10-7 (6.74· 10-7)100%
2 Layer Transformer1.97·10-2 (2.41·10-299.6%4.63· 10-2 (6.72 : 10-298.7%
P=535.96·10-5 (8.91·10-5)100%1.5·10-4 (2.70· 10-4)100%
P=1091.94· 10-7 (3.74· 10-8)100%6.53· 10-7 (1.41 · 10-7)100%
Dropout p = 0.20.215 (0.091)99.7%0.205 (0.075)99.7%
Dropout p = 0.54.68 · 10-³ (8.11· 10-3)100%3.6:10-3 (5.82.10-3)100%
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Note", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "that training and test loss coincide, as the model does not see repeated pairs.(Bottom) The train/test", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "score": 1.0, + "content": "loss per token for 5 Digit Addition, trained with randomly generated data at each step. Note that", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "phase changes in the average loss correspond to phase changes in individual tokens, though one", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "phase change (token 1, around step 270) is not visible on the averaged loss as it overlaps with the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 445, + 365, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 365, + 457 + ], + "score": 1.0, + "content": "end of the first phase change (token 0, starting around step 150).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 486, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 105, + 485, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 499 + ], + "score": 1.0, + "content": "optimizers. We were even able to induce slingshots on a 1-layer by reducing the precision of the loss", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "calculations (as this causes many gradients to round to 0 and thus greatly increases the differences", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 508, + 195, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 195, + 520 + ], + "score": 1.0, + "content": "in scale of gradients).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 524, + 505, + 580 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "However, as many of our 1-layer models do not exhibit slingshots but nonetheless grok, the slingshot", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 536, + 504, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 504, + 548 + ], + "score": 1.0, + "content": "mechanism is unnecessary for grokking to occur, in the presence of weight decay or other regular-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "ization. We speculate that the slingshots of Thilak et al. (2022) (which co-occur with grokking for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "score": 1.0, + "content": "training runs without weight decay) serve as an implicit regularization mechanism that favors the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 568, + 336, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 336, + 581 + ], + "score": 1.0, + "content": "simpler, generalizing solution over the more complicated", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "title", + "bbox": [ + 107, + 603, + 387, + 614 + ], + "lines": [ + { + "bbox": [ + 106, + 603, + 388, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 388, + 615 + ], + "score": 1.0, + "content": "D.3 ADDITIONAL EVIDENCE FROM OTHER ALGORITHMIC TASKS", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "We now provide addition analysis of grokking phenomena on 3 additional algorithmic tasks and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 636, + 339, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 339, + 652 + ], + "score": 1.0, + "content": "confirm that limited data is an important part of grokking:", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 127, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 129, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 129, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "1. 5 digit addition. We sample pairs of random 5 digit numbers and have the model predict", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 141, + 666, + 183, + 678 + ], + "spans": [ + { + "bbox": [ + 141, + 666, + 183, + 678 + ], + "score": 1.0, + "content": "their sum", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 128, + 676, + 504, + 690 + ], + "spans": [ + { + "bbox": [ + 128, + 676, + 504, + 690 + ], + "score": 1.0, + "content": "2. Predicting repeated subsequences. We take a uniform random sequence of tokens, ran-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 141, + 688, + 501, + 700 + ], + "spans": [ + { + "bbox": [ + 141, + 688, + 501, + 700 + ], + "score": 1.0, + "content": "domly choose a subsequence to repeat, and train the model to predict the repeated tokens.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 128, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 128, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "3. Skip trigram. We feed in a sequence of tokens from 0 to 19, of which exactly one is", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 141, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 141, + 710, + 441, + 722 + ], + "score": 1.0, + "content": "greater than or equal to 10, and the model needs to output the token that is", + "type": "text" + }, + { + "bbox": [ + 442, + 710, + 464, + 721 + ], + "score": 0.87, + "content": "\\geq 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 710, + 505, + 722 + ], + "score": 1.0, + "content": ". This can", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 142, + 720, + 307, + 733 + ], + "spans": [ + { + "bbox": [ + 142, + 720, + 307, + 733 + ], + "score": 1.0, + "content": "be solved with learning 10 skip trigrams.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42 + } + ], + "page_idx": 29, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "30", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 181, + 81, + 427, + 374 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 181, + 81, + 427, + 374 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 181, + 81, + 427, + 374 + ], + "spans": [ + { + "bbox": [ + 181, + 81, + 427, + 374 + ], + "score": 0.964, + "type": "image", + "image_path": "98e36469ef9a987a61c0b01f800e8d97e6a844a791ccd5e6b0e1101470a9d906.jpg" + } + ] + } + ], + "index": 10.5, + "virtual_lines": [ + { + "bbox": [ + 181, + 81, + 427, + 94.31818181818181 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 181, + 94.31818181818181, + 427, + 107.63636363636363 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 181, + 107.63636363636363, + 427, + 120.95454545454544 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 181, + 120.95454545454544, + 427, + 134.27272727272725 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 181, + 134.27272727272725, + 427, + 147.59090909090907 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 181, + 147.59090909090907, + 427, + 160.90909090909088 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 181, + 160.90909090909088, + 427, + 174.2272727272727 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 181, + 174.2272727272727, + 427, + 187.5454545454545 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 181, + 187.5454545454545, + 427, + 200.86363636363632 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 181, + 200.86363636363632, + 427, + 214.18181818181813 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 181, + 214.18181818181813, + 427, + 227.49999999999994 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 181, + 227.49999999999994, + 427, + 240.81818181818176 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 181, + 240.81818181818176, + 427, + 254.13636363636357 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 181, + 254.13636363636357, + 427, + 267.4545454545454 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 181, + 267.4545454545454, + 427, + 280.7727272727272 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 181, + 280.7727272727272, + 427, + 294.090909090909 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 181, + 294.090909090909, + 427, + 307.4090909090908 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 181, + 307.4090909090908, + 427, + 320.72727272727263 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 181, + 320.72727272727263, + 427, + 334.04545454545445 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 181, + 334.04545454545445, + 427, + 347.36363636363626 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 181, + 347.36363636363626, + 427, + 360.6818181818181 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 181, + 360.6818181818181, + 427, + 373.9999999999999 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 389, + 505, + 456 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "Figure 29: (Top) The training/test loss for 5 Digit Addition trained on randomly generated data. Note", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "that training and test loss coincide, as the model does not see repeated pairs.(Bottom) The train/test", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "score": 1.0, + "content": "loss per token for 5 Digit Addition, trained with randomly generated data at each step. Note that", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "phase changes in the average loss correspond to phase changes in individual tokens, though one", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "phase change (token 1, around step 270) is not visible on the averaged loss as it overlaps with the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 445, + 365, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 365, + 457 + ], + "score": 1.0, + "content": "end of the first phase change (token 0, starting around step 150).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 486, + 505, + 519 + ], + "lines": [], + "index": 29, + "bbox_fs": [ + 105, + 485, + 505, + 520 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 524, + 505, + 580 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "However, as many of our 1-layer models do not exhibit slingshots but nonetheless grok, the slingshot", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 536, + 504, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 504, + 548 + ], + "score": 1.0, + "content": "mechanism is unnecessary for grokking to occur, in the presence of weight decay or other regular-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "ization. We speculate that the slingshots of Thilak et al. (2022) (which co-occur with grokking for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "score": 1.0, + "content": "training runs without weight decay) serve as an implicit regularization mechanism that favors the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 568, + 336, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 336, + 581 + ], + "score": 1.0, + "content": "simpler, generalizing solution over the more complicated", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 525, + 505, + 581 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 603, + 387, + 614 + ], + "lines": [ + { + "bbox": [ + 106, + 603, + 388, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 388, + 615 + ], + "score": 1.0, + "content": "D.3 ADDITIONAL EVIDENCE FROM OTHER ALGORITHMIC TASKS", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "We now provide addition analysis of grokking phenomena on 3 additional algorithmic tasks and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 636, + 339, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 339, + 652 + ], + "score": 1.0, + "content": "confirm that limited data is an important part of grokking:", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 626, + 505, + 652 + ] + }, + { + "type": "list", + "bbox": [ + 127, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 129, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 129, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "1. 5 digit addition. We sample pairs of random 5 digit numbers and have the model predict", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 666, + 183, + 678 + ], + "spans": [ + { + "bbox": [ + 141, + 666, + 183, + 678 + ], + "score": 1.0, + "content": "their sum", + "type": "text" + } + ], + "index": 40, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 676, + 504, + 690 + ], + "spans": [ + { + "bbox": [ + 128, + 676, + 504, + 690 + ], + "score": 1.0, + "content": "2. Predicting repeated subsequences. We take a uniform random sequence of tokens, ran-", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 688, + 501, + 700 + ], + "spans": [ + { + "bbox": [ + 141, + 688, + 501, + 700 + ], + "score": 1.0, + "content": "domly choose a subsequence to repeat, and train the model to predict the repeated tokens.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 128, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 128, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "3. Skip trigram. We feed in a sequence of tokens from 0 to 19, of which exactly one is", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 141, + 710, + 441, + 722 + ], + "score": 1.0, + "content": "greater than or equal to 10, and the model needs to output the token that is", + "type": "text" + }, + { + "bbox": [ + 442, + 710, + 464, + 721 + ], + "score": 0.87, + "content": "\\geq 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 710, + 505, + 722 + ], + "score": 1.0, + "content": ". This can", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 142, + 720, + 307, + 733 + ], + "spans": [ + { + "bbox": [ + 142, + 720, + 307, + 733 + ], + "score": 1.0, + "content": "be solved with learning 10 skip trigrams.", + "type": "text" + } + ], + "index": 45, + "is_list_end_line": true + } + ], + "index": 42, + "bbox_fs": [ + 128, + 654, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 194, + 111, + 402, + 223 + ], + "blocks": [ + { + "type": "image_caption", + "bbox": [ + 201, + 100, + 410, + 110 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 201, + 99, + 410, + 111 + ], + "spans": [ + { + "bbox": [ + 201, + 99, + 410, + 111 + ], + "score": 1.0, + "content": "Train/Test Loss for 5 Digit Addition with 700 Data Points", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "image_body", + "bbox": [ + 194, + 111, + 402, + 223 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 194, + 111, + 402, + 223 + ], + "spans": [ + { + "bbox": [ + 194, + 111, + 402, + 223 + ], + "score": 0.949, + "type": "image", + "image_path": "22f8891022c060decc4a3a6f42852fdbbd2ad156eb7aa23bd0a93328710333b7.jpg" + } + ] + } + ], + "index": 4.5, + "virtual_lines": [ + { + "bbox": [ + 194, + 111, + 402, + 125.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 194, + 125.0, + 402, + 139.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 194, + 139.0, + 402, + 153.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 194, + 153.0, + 402, + 167.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 194, + 167.0, + 402, + 181.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 194, + 181.0, + 402, + 195.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 194, + 195.0, + 402, + 209.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 194, + 209.0, + 402, + 223.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 238, + 504, + 261 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 238, + 504, + 251 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 504, + 251 + ], + "score": 1.0, + "content": "Figure 30: The train and test loss for 5 Digit Addition trained on 700 data points. Unlike the infinite,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 249, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 505, + 263 + ], + "score": 1.0, + "content": "randomly generated data case, this shows both a sharp phase change and clear train test divergence.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 282, + 504, + 316 + ], + "lines": [ + { + "bbox": [ + 105, + 282, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 505, + 296 + ], + "score": 1.0, + "content": "We use a 1-layer full transformer for 5-digit addition, a 2-layer attention only transformer for pre-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 294, + 504, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 504, + 306 + ], + "score": 1.0, + "content": "dicting repeated subsequences, and a 1-layer attention only transformer for the skip trigram task.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 304, + 392, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 392, + 318 + ], + "score": 1.0, + "content": "Otherwise, we use the same hyperparameters as in the mainline model.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 329, + 505, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "5 Digit Addition We first consider the case where we train on the approximately infinite data", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "regime. For each minibatch, we randomly new sample 5 digit numbers. We report the results in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "score": 1.0, + "content": "Figure 29. Train loss coincides with test loss, so grokking does not occur, as the model almost never", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 309, + 375 + ], + "score": 1.0, + "content": "sees the same pair of 5 digit numbers twice, with", + "type": "text" + }, + { + "bbox": [ + 309, + 361, + 329, + 372 + ], + "score": 0.87, + "content": "1 0 ^ { 1 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 361, + 506, + 375 + ], + "score": 1.0, + "content": "such pairs. Interestingly, the various small", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "bumps in Figure 29 correspond to the model learning how to calculate each of the 6 tokens in the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "output. However, grokking does occur when we restrict the model to only see 700 data points, as", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 395, + 188, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 188, + 407 + ], + "score": 1.0, + "content": "shown in Figure 30.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 418, + 505, + 463 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "score": 1.0, + "content": "Repeated subsequence As with the 5-digit addition task, we find that restricting the amount of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "data is necessary and sufficient for grokking on the repeated subsequence task. In Figure 31, the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "score": 1.0, + "content": "model sees new data at every step exhibits no grokking. In contrast, clear grokking occurs when we", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 452, + 341, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 341, + 464 + ], + "score": 1.0, + "content": "restrict the model to only see 512 data points in Figure 32.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 475, + 505, + 520 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "Skip trigram As with the previous tasks, we find that restricting the amount of data is necessary", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "and sufficient for grokking on the skip trigram task. The model that sees new data at every step", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "exhibits no grokking in Figure 33. Meanwhile, the model restricted to only see 512 data points", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 508, + 254, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 254, + 521 + ], + "score": 1.0, + "content": "exhibits clear grokking in Figure 34.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 108, + 525, + 419, + 537 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 421, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 421, + 540 + ], + "score": 1.0, + "content": "Taken together, these results echo the importance of limited data for grokking.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 107, + 553, + 332, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 334, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 334, + 568 + ], + "score": 1.0, + "content": "E FURTHER SPECULATIONS ON GROKKING", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 109, + 578, + 318, + 589 + ], + "lines": [ + { + "bbox": [ + 106, + 578, + 320, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 320, + 590 + ], + "score": 1.0, + "content": "E.1 AN INTUITIVE EXPLANATION OF GROKKING", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "score": 1.0, + "content": "In this section, we speculate on what might be happening “under the hood” when a model groks and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "explore why this phenomena happens. The evidence is only suggestive, so this a promising direction", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 621, + 185, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 185, + 633 + ], + "score": 1.0, + "content": "for future research.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 638, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "Grokking occurs when models, trained on algorithmic tasks with certain hyperparameters, initially", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "overfit the training data where train loss significantly improves while test loss worsens and the two", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "diverge. But later in training, there is a sudden improvement in test loss, so test and train loss", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "converge. In contrast to Power et al. (2022) but in line with Liu et al. (2022), grokking does not", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "occur when both train and test loss improve together without the initial divergence, as shown in", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 693, + 362, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 362, + 705 + ], + "score": 1.0, + "content": "many of the figures in this paper, for example Figures 2 and 18.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "The core issue is that the model has two possible solutions: memorization (with low train loss", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 721, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 107, + 721, + 504, + 732 + ], + "score": 1.0, + "content": "and high test loss) and a generalization (with low train loss and low test loss). In our case, the", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5 + } + ], + "page_idx": 30, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 194, + 111, + 402, + 223 + ], + "blocks": [ + { + "type": "image_caption", + "bbox": [ + 201, + 100, + 410, + 110 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 201, + 99, + 410, + 111 + ], + "spans": [ + { + "bbox": [ + 201, + 99, + 410, + 111 + ], + "score": 1.0, + "content": "Train/Test Loss for 5 Digit Addition with 700 Data Points", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "image_body", + "bbox": [ + 194, + 111, + 402, + 223 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 194, + 111, + 402, + 223 + ], + "spans": [ + { + "bbox": [ + 194, + 111, + 402, + 223 + ], + "score": 0.949, + "type": "image", + "image_path": "22f8891022c060decc4a3a6f42852fdbbd2ad156eb7aa23bd0a93328710333b7.jpg" + } + ] + } + ], + "index": 4.5, + "virtual_lines": [ + { + "bbox": [ + 194, + 111, + 402, + 125.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 194, + 125.0, + 402, + 139.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 194, + 139.0, + 402, + 153.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 194, + 153.0, + 402, + 167.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 194, + 167.0, + 402, + 181.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 194, + 181.0, + 402, + 195.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 194, + 195.0, + 402, + 209.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 194, + 209.0, + 402, + 223.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 238, + 504, + 261 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 238, + 504, + 251 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 504, + 251 + ], + "score": 1.0, + "content": "Figure 30: The train and test loss for 5 Digit Addition trained on 700 data points. Unlike the infinite,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 249, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 505, + 263 + ], + "score": 1.0, + "content": "randomly generated data case, this shows both a sharp phase change and clear train test divergence.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 282, + 504, + 316 + ], + "lines": [ + { + "bbox": [ + 105, + 282, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 505, + 296 + ], + "score": 1.0, + "content": "We use a 1-layer full transformer for 5-digit addition, a 2-layer attention only transformer for pre-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 294, + 504, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 504, + 306 + ], + "score": 1.0, + "content": "dicting repeated subsequences, and a 1-layer attention only transformer for the skip trigram task.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 304, + 392, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 392, + 318 + ], + "score": 1.0, + "content": "Otherwise, we use the same hyperparameters as in the mainline model.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 282, + 505, + 318 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 329, + 505, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "5 Digit Addition We first consider the case where we train on the approximately infinite data", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "regime. For each minibatch, we randomly new sample 5 digit numbers. We report the results in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "score": 1.0, + "content": "Figure 29. Train loss coincides with test loss, so grokking does not occur, as the model almost never", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 309, + 375 + ], + "score": 1.0, + "content": "sees the same pair of 5 digit numbers twice, with", + "type": "text" + }, + { + "bbox": [ + 309, + 361, + 329, + 372 + ], + "score": 0.87, + "content": "1 0 ^ { 1 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 361, + 506, + 375 + ], + "score": 1.0, + "content": "such pairs. Interestingly, the various small", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "bumps in Figure 29 correspond to the model learning how to calculate each of the 6 tokens in the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "output. However, grokking does occur when we restrict the model to only see 700 data points, as", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 395, + 188, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 188, + 407 + ], + "score": 1.0, + "content": "shown in Figure 30.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 329, + 506, + 407 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 418, + 505, + 463 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "score": 1.0, + "content": "Repeated subsequence As with the 5-digit addition task, we find that restricting the amount of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "data is necessary and sufficient for grokking on the repeated subsequence task. In Figure 31, the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "score": 1.0, + "content": "model sees new data at every step exhibits no grokking. In contrast, clear grokking occurs when we", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 452, + 341, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 341, + 464 + ], + "score": 1.0, + "content": "restrict the model to only see 512 data points in Figure 32.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 419, + 505, + 464 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 475, + 505, + 520 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "Skip trigram As with the previous tasks, we find that restricting the amount of data is necessary", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "and sufficient for grokking on the skip trigram task. The model that sees new data at every step", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "exhibits no grokking in Figure 33. Meanwhile, the model restricted to only see 512 data points", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 508, + 254, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 254, + 521 + ], + "score": 1.0, + "content": "exhibits clear grokking in Figure 34.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 475, + 505, + 521 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 525, + 419, + 537 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 421, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 421, + 540 + ], + "score": 1.0, + "content": "Taken together, these results echo the importance of limited data for grokking.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 522, + 421, + 540 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 553, + 332, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 334, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 334, + 568 + ], + "score": 1.0, + "content": "E FURTHER SPECULATIONS ON GROKKING", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 109, + 578, + 318, + 589 + ], + "lines": [ + { + "bbox": [ + 106, + 578, + 320, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 320, + 590 + ], + "score": 1.0, + "content": "E.1 AN INTUITIVE EXPLANATION OF GROKKING", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "score": 1.0, + "content": "In this section, we speculate on what might be happening “under the hood” when a model groks and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "explore why this phenomena happens. The evidence is only suggestive, so this a promising direction", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 621, + 185, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 185, + 633 + ], + "score": 1.0, + "content": "for future research.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 598, + 505, + 633 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 638, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "Grokking occurs when models, trained on algorithmic tasks with certain hyperparameters, initially", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "overfit the training data where train loss significantly improves while test loss worsens and the two", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "diverge. But later in training, there is a sudden improvement in test loss, so test and train loss", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "converge. In contrast to Power et al. (2022) but in line with Liu et al. (2022), grokking does not", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "occur when both train and test loss improve together without the initial divergence, as shown in", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 693, + 362, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 362, + 705 + ], + "score": 1.0, + "content": "many of the figures in this paper, for example Figures 2 and 18.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 637, + 505, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "The core issue is that the model has two possible solutions: memorization (with low train loss", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 721, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 107, + 721, + 504, + 732 + ], + "score": 1.0, + "content": "and high test loss) and a generalization (with low train loss and low test loss). 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(2022) and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 412, + 504, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 504, + 424 + ], + "score": 1.0, + "content": "our results in Appendix D.3. That is, rather than slowly forming that capability over training, the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "model rapidly goes from being bad at it to being good at it. One interpretation of a phase transition", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "is that there’s some feature of the loss landscape that makes the generalising solution harder to reach", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "- rather than a smooth gradient for the model to follow, it instead initially finds it difficult to make", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 456, + 419, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 419, + 469 + ], + "score": 1.0, + "content": "progress, but then crosses some threshold where it can rapidly make progress.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 473, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "Therefore, grokking occurs with phase transitions, limited data, and regularization. Models exhibit", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "phase transitions despite having enough training data to avoid overfitting. Regularization (weight", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "score": 1.0, + "content": "decay in our case) favors simpler solutions over complex ones. The model has enough data to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 506, + 504, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 504, + 518 + ], + "score": 1.0, + "content": "marginally prefer generalization over memorization. The phase transition indicates that generaliza-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 516, + 504, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 504, + 528 + ], + "score": 1.0, + "content": "tion is “hard to reach” while the model has no problems with memorization. But as it memorizes, the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "network becomes more complex until the weight decay prevents further memorization then moves", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "towards equilibrium. The gradient to memorize balances the gradient towards smaller weights. With", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 549, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 562 + ], + "score": 1.0, + "content": "generalization, the model is incentivized to both memorize and simplify. Strikingly, it is capable of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "both while maintaining a somewhat constant training performance in this circuit formation phase.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 570, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 586 + ], + "score": 1.0, + "content": "Next, as the model approaches generalization, the memorization weights are removed in the cleanup", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 582, + 504, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 504, + 594 + ], + "score": 1.0, + "content": "phase. The cost from complexity outweighs the benefit from lower loss. Due to the phase transition", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "score": 1.0, + "content": "during this training period, as model’s progress towards generalization accelerates, the cleanup rate", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 604, + 177, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 177, + 617 + ], + "score": 1.0, + "content": "sharpens as well.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 505, + 698 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "score": 1.0, + "content": "A model that learns a perfect solution and is trained with weight decay has competing incentives:", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "larger weights (for more extreme logits and thus lower loss) and smaller weights (from weight", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "decay). So for any solution and any level of weight decay, there will always be a level of train loss", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 653, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 506, + 667 + ], + "score": 1.0, + "content": "where these two forces equilibrate. Thus, memorization is not necessarily a “simpler” solution than", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 664, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 678 + ], + "score": 1.0, + "content": "generalization. The key is that generalization will have smaller weights holding train loss fixed. In", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "fact, weight decay should be expected to equilibrate at a slightly lower train loss in generalization,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 687, + 420, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 420, + 699 + ], + "score": 1.0, + "content": "since the base solution is simpler. This matches what we observe in practice. 4", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43 + } + ], + "page_idx": 32, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 712, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 119, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "4One subtlety: the grokking phenomena is often incorrectly summarized as “the model learned to generalize", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "even after achieving zero loss.” Zero loss does not exist with cross-entropy loss. 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Unlike the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "score": 1.0, + "content": "infinite, randomly generated data case, this shows both a sharp phase change and clear train test", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 259, + 156, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 156, + 275 + ], + "score": 1.0, + "content": "divergence.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 296, + 505, + 373 + ], + "lines": [ + { + "bbox": [ + 105, + 295, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 310 + ], + "score": 1.0, + "content": "Fourier Multiplication Algorithm is the generalization solution. Intuitively, with very little training", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 308, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 319 + ], + "score": 1.0, + "content": "data, the model will overfit and memorize. With more training data, the model must generalize", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "or suffer poor performance on both train and test loss. Since neural networks have an inductive", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "bias favoring “simpler” solutions, memorization complexity scales with the size of the training set,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 339, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 505, + 354 + ], + "score": 1.0, + "content": "whereas generalization complexity is constant. The two must cross at some point! Yet, the surprising", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 351, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 364 + ], + "score": 1.0, + "content": "aspect of grokking is the abrupt shift during training, when the model switches from memorization", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 362, + 178, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 178, + 374 + ], + "score": 1.0, + "content": "to generalization.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 295, + 506, + 374 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 505, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "score": 1.0, + "content": "The other component of grokking is phase transitions - the phenomena where models trained on a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 391, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 505, + 402 + ], + "score": 1.0, + "content": "certain task develop a specific capability fairly rapidly during a brief period of training, as shown", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 400, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 413 + ], + "score": 1.0, + "content": "for the case of induction heads forming in transformer language models in Olsson et al. (2022) and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 412, + 504, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 504, + 424 + ], + "score": 1.0, + "content": "our results in Appendix D.3. That is, rather than slowly forming that capability over training, the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "model rapidly goes from being bad at it to being good at it. One interpretation of a phase transition", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "is that there’s some feature of the loss landscape that makes the generalising solution harder to reach", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "- rather than a smooth gradient for the model to follow, it instead initially finds it difficult to make", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 456, + 419, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 419, + 469 + ], + "score": 1.0, + "content": "progress, but then crosses some threshold where it can rapidly make progress.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 378, + 506, + 469 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 473, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "Therefore, grokking occurs with phase transitions, limited data, and regularization. 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Strikingly, it is capable of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "both while maintaining a somewhat constant training performance in this circuit formation phase.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 570, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 586 + ], + "score": 1.0, + "content": "Next, as the model approaches generalization, the memorization weights are removed in the cleanup", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 582, + 504, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 504, + 594 + ], + "score": 1.0, + "content": "phase. The cost from complexity outweighs the benefit from lower loss. Due to the phase transition", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "score": 1.0, + "content": "during this training period, as model’s progress towards generalization accelerates, the cleanup rate", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 604, + 177, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 177, + 617 + ], + "score": 1.0, + "content": "sharpens as well.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 473, + 506, + 617 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 505, + 698 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "score": 1.0, + "content": "A model that learns a perfect solution and is trained with weight decay has competing incentives:", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "larger weights (for more extreme logits and thus lower loss) and smaller weights (from weight", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "decay). So for any solution and any level of weight decay, there will always be a level of train loss", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 653, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 506, + 667 + ], + "score": 1.0, + "content": "where these two forces equilibrate. Thus, memorization is not necessarily a “simpler” solution than", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 664, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 678 + ], + "score": 1.0, + "content": "generalization. The key is that generalization will have smaller weights holding train loss fixed. In", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "fact, weight decay should be expected to equilibrate at a slightly lower train loss in generalization,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 687, + 420, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 420, + 699 + ], + "score": 1.0, + "content": "since the base solution is simpler. This matches what we observe in practice. 4", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 621, + 506, + 699 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 426, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 428, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 428, + 95 + ], + "score": 1.0, + "content": "E.2 HYPOTHESIS: PHASE TRANSITIONS ARE INHERENT TO COMPOSITION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 102, + 505, + 191 + ], + "lines": [ + { + "bbox": [ + 105, + 103, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 115 + ], + "score": 1.0, + "content": "A promising line of work in the growing field of mechanistic interpretability suggests that models", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 115, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 126 + ], + "score": 1.0, + "content": "form circuits (Cammarata et al., 2020) – clean interpretable algorithms formed by subnetworks of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 125, + 505, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 505, + 137 + ], + "score": 1.0, + "content": "the model, such as curve detectors (Cammarata et al., 2020) in image classification networks and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 505, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 148 + ], + "score": 1.0, + "content": "induction heads (Elhage et al., 2021; Olsson et al., 2022) in LLMs. This is surprisingly true! A", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 147, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 159 + ], + "score": 1.0, + "content": "circuit represents the model learning an algorithm, a fundamentally discrete thing; each step in the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 157, + 504, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 157, + 504, + 169 + ], + "score": 1.0, + "content": "algorithm only makes sense if the other steps are present. But neural networks are fundamentally", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 169, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 505, + 181 + ], + "score": 1.0, + "content": "continuous, trained to follow gradients towards lower loss and struggle to jump to new optima with-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 180, + 432, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 432, + 192 + ], + "score": 1.0, + "content": "out following a smooth gradient. So how can a model learn a discrete algorithm?", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 196, + 505, + 306 + ], + "lines": [ + { + "bbox": [ + 106, + 197, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 505, + 208 + ], + "score": 1.0, + "content": "As a concrete example, let’s consider the case of induction heads in LLMs. There is a subnetwork", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 209, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 220 + ], + "score": 1.0, + "content": "of a next-token prediction autoregressive language model that learns to continue repeated subse-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "score": 1.0, + "content": "quences. It detects whether the current token occurred earlier in the context. If so, it predicts the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 230, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 505, + 242 + ], + "score": 1.0, + "content": "same token after that previous occurrence will also come next. The circuit consists of a previous", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "score": 1.0, + "content": "token head, which attends to each previous token and copies the context of the previous token to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 252, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 505, + 263 + ], + "score": 1.0, + "content": "the current token, and an induction head which attends to the token after a previous occurrence of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "score": 1.0, + "content": "the current token. The induction head composes with the previous token head by forming a query", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 274, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 505, + 285 + ], + "score": 1.0, + "content": "vector representing the current token and a key vector representing the previous token head’s output", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 284, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 506, + 297 + ], + "score": 1.0, + "content": "using K-Composition, the context of the previous token. It attends to a token where this query and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 295, + 153, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 153, + 308 + ], + "score": 1.0, + "content": "key match.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 312, + 505, + 389 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "This circuit significantly improves loss but only in the context of the other heads present. Before", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "score": 1.0, + "content": "either head is present, no gradient encourages the formation of either head. At initialization, we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "have neither head, so gradient descent should never discover this circuit. Naively, we might predict", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "score": 1.0, + "content": "that neural networks will only produce circuits analogous to linear regression, where each weight", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 357, + 504, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 504, + 368 + ], + "score": 1.0, + "content": "will marginally improve performance as it continuously trains. And yet in practice, neural networks", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "score": 1.0, + "content": "indeed form such sophisticated circuits, involving several parts interacting in non-trivial, algorithmic", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 378, + 214, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 214, + 390 + ], + "score": 1.0, + "content": "ways. So how can this be?", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 108, + 395, + 222, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 394, + 224, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 224, + 408 + ], + "score": 1.0, + "content": "A few possible explanations:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 133, + 415, + 504, + 503 + ], + "lines": [ + { + "bbox": [ + 133, + 414, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 133, + 414, + 505, + 427 + ], + "score": 1.0, + "content": "• Lottery tickets (Frankle & Carbin, 2018): Initially, each layer of the network is the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 426, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 141, + 426, + 506, + 439 + ], + "score": 1.0, + "content": "superposition of many partial circuit components, and the output of each layer is the average", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 141, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "of the output of each component. The full output of the network is the average of many", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 141, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "different circuits, with significant interference from non-linear interaction. Some of these", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 142, + 459, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 142, + 459, + 505, + 471 + ], + "score": 1.0, + "content": "circuits are systematically useful to reducing loss, but most aren’t. Gradients for useless", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 469, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 141, + 469, + 505, + 483 + ], + "score": 1.0, + "content": "circuits will have zero mean, while gradients for useful circuits will have non-zero mean,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 141, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "with a lot of noise. SGD reinforces relevant circuits and suppresses useless ones, so circuits", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 142, + 492, + 225, + 505 + ], + "spans": [ + { + "bbox": [ + 142, + 492, + 225, + 505 + ], + "score": 1.0, + "content": "will gradually form.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 135, + 507, + 503, + 550 + ], + "lines": [ + { + "bbox": [ + 133, + 506, + 504, + 519 + ], + "spans": [ + { + "bbox": [ + 133, + 506, + 504, + 519 + ], + "score": 1.0, + "content": "• Random walk: The network wanders randomly around the loss landscape until it encoun-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 518, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 141, + 518, + 505, + 529 + ], + "score": 1.0, + "content": "ters a half-formed previous token head and induction head that somewhat compose. This", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 141, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "half-formed circuit becomes useful for reducing loss, so gradient descent completes the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 540, + 173, + 551 + ], + "spans": [ + { + "bbox": [ + 141, + 540, + 173, + 551 + ], + "score": 1.0, + "content": "circuit.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 134, + 554, + 504, + 609 + ], + "lines": [ + { + "bbox": [ + 133, + 553, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 133, + 553, + 505, + 567 + ], + "score": 1.0, + "content": "• Evolution: A similar mystery arises from how organisms develop sophisticated machinery,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 141, + 564, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 141, + 564, + 505, + 578 + ], + "score": 1.0, + "content": "like the human eye. Each part is only useful in the context of other parts. A compelling", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 141, + 576, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 141, + 576, + 505, + 587 + ], + "score": 1.0, + "content": "explanation is a component first developed that was somewhat useful in its own right, like", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 141, + 586, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 141, + 586, + 505, + 600 + ], + "score": 1.0, + "content": "a light-detecting membrane. It was reinforced as a useful component. Then, later compo-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 141, + 598, + 396, + 611 + ], + "spans": [ + { + "bbox": [ + 141, + 598, + 396, + 611 + ], + "score": 1.0, + "content": "nents developed depending on the first, like the lens of the eye.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 106, + 618, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 617, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 631 + ], + "score": 1.0, + "content": "Evolution is a natural explanation, However, based on our toy tasks, it cannot be the whole story.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "In the repeated subsequence task, we have a sequence of uniform randomly generated tokens, apart", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "from a repeated subsequence at an arbitrary location, e.g. 7 2 8 3 1 9 3 8 3 1 9 9 2 5 END. This means", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "all pairs of tokens are independent, apart from pairs of equal tokens in the repeated subsequence. In", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 662, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 505, + 673 + ], + "score": 1.0, + "content": "particular, this means that a previous token head can never reduce loss for the current token. The", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "previous token will always be independent of the next token. So a previous token head is only useful", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 684, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 684, + 505, + 695 + ], + "score": 1.0, + "content": "in the context of an induction-like head that completes the circuit. Likewise, an induction head relies", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47 + } + ], + "page_idx": 33, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 702, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 701, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 506, + 713 + ], + "score": 1.0, + "content": "perfect accuracy, it is trained to optimize loss not accuracy. This means the model is always incentivized to", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 710, + 506, + 725 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 725 + ], + "score": 1.0, + "content": "further improve. In particular, the easiest way to improve performance with perfect accuracy is by scaling up", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 408, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 408, + 733 + ], + "score": 1.0, + "content": "the logits. This lowers the temperature and pushes the softmax closer to an argmax.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 752, + 311, + 760 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 426, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 428, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 428, + 95 + ], + "score": 1.0, + "content": "E.2 HYPOTHESIS: PHASE TRANSITIONS ARE INHERENT TO COMPOSITION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 82, + 428, + 95 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 102, + 505, + 191 + ], + "lines": [ + { + "bbox": [ + 105, + 103, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 115 + ], + "score": 1.0, + "content": "A promising line of work in the growing field of mechanistic interpretability suggests that models", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 115, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 126 + ], + "score": 1.0, + "content": "form circuits (Cammarata et al., 2020) – clean interpretable algorithms formed by subnetworks of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 125, + 505, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 505, + 137 + ], + "score": 1.0, + "content": "the model, such as curve detectors (Cammarata et al., 2020) in image classification networks and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 505, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 148 + ], + "score": 1.0, + "content": "induction heads (Elhage et al., 2021; Olsson et al., 2022) in LLMs. This is surprisingly true! A", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 147, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 159 + ], + "score": 1.0, + "content": "circuit represents the model learning an algorithm, a fundamentally discrete thing; each step in the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 157, + 504, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 157, + 504, + 169 + ], + "score": 1.0, + "content": "algorithm only makes sense if the other steps are present. But neural networks are fundamentally", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 169, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 505, + 181 + ], + "score": 1.0, + "content": "continuous, trained to follow gradients towards lower loss and struggle to jump to new optima with-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 180, + 432, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 432, + 192 + ], + "score": 1.0, + "content": "out following a smooth gradient. So how can a model learn a discrete algorithm?", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 103, + 505, + 192 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 196, + 505, + 306 + ], + "lines": [ + { + "bbox": [ + 106, + 197, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 505, + 208 + ], + "score": 1.0, + "content": "As a concrete example, let’s consider the case of induction heads in LLMs. There is a subnetwork", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 209, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 220 + ], + "score": 1.0, + "content": "of a next-token prediction autoregressive language model that learns to continue repeated subse-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "score": 1.0, + "content": "quences. It detects whether the current token occurred earlier in the context. If so, it predicts the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 230, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 505, + 242 + ], + "score": 1.0, + "content": "same token after that previous occurrence will also come next. The circuit consists of a previous", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "score": 1.0, + "content": "token head, which attends to each previous token and copies the context of the previous token to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 252, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 505, + 263 + ], + "score": 1.0, + "content": "the current token, and an induction head which attends to the token after a previous occurrence of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "score": 1.0, + "content": "the current token. The induction head composes with the previous token head by forming a query", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 274, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 505, + 285 + ], + "score": 1.0, + "content": "vector representing the current token and a key vector representing the previous token head’s output", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 284, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 506, + 297 + ], + "score": 1.0, + "content": "using K-Composition, the context of the previous token. It attends to a token where this query and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 295, + 153, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 153, + 308 + ], + "score": 1.0, + "content": "key match.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 197, + 506, + 308 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 312, + 505, + 389 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "This circuit significantly improves loss but only in the context of the other heads present. Before", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "score": 1.0, + "content": "either head is present, no gradient encourages the formation of either head. At initialization, we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "have neither head, so gradient descent should never discover this circuit. Naively, we might predict", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "score": 1.0, + "content": "that neural networks will only produce circuits analogous to linear regression, where each weight", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 357, + 504, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 504, + 368 + ], + "score": 1.0, + "content": "will marginally improve performance as it continuously trains. And yet in practice, neural networks", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "score": 1.0, + "content": "indeed form such sophisticated circuits, involving several parts interacting in non-trivial, algorithmic", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 378, + 214, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 214, + 390 + ], + "score": 1.0, + "content": "ways. So how can this be?", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 312, + 506, + 390 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 395, + 222, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 394, + 224, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 224, + 408 + ], + "score": 1.0, + "content": "A few possible explanations:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 394, + 224, + 408 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 415, + 504, + 503 + ], + "lines": [ + { + "bbox": [ + 133, + 414, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 133, + 414, + 505, + 427 + ], + "score": 1.0, + "content": "• Lottery tickets (Frankle & Carbin, 2018): Initially, each layer of the network is the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 426, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 141, + 426, + 506, + 439 + ], + "score": 1.0, + "content": "superposition of many partial circuit components, and the output of each layer is the average", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 141, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "of the output of each component. The full output of the network is the average of many", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 141, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "different circuits, with significant interference from non-linear interaction. Some of these", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 142, + 459, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 142, + 459, + 505, + 471 + ], + "score": 1.0, + "content": "circuits are systematically useful to reducing loss, but most aren’t. Gradients for useless", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 469, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 141, + 469, + 505, + 483 + ], + "score": 1.0, + "content": "circuits will have zero mean, while gradients for useful circuits will have non-zero mean,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 141, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "with a lot of noise. SGD reinforces relevant circuits and suppresses useless ones, so circuits", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 142, + 492, + 225, + 505 + ], + "spans": [ + { + "bbox": [ + 142, + 492, + 225, + 505 + ], + "score": 1.0, + "content": "will gradually form.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 133, + 414, + 506, + 505 + ] + }, + { + "type": "text", + "bbox": [ + 135, + 507, + 503, + 550 + ], + "lines": [ + { + "bbox": [ + 133, + 506, + 504, + 519 + ], + "spans": [ + { + "bbox": [ + 133, + 506, + 504, + 519 + ], + "score": 1.0, + "content": "• Random walk: The network wanders randomly around the loss landscape until it encoun-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 518, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 141, + 518, + 505, + 529 + ], + "score": 1.0, + "content": "ters a half-formed previous token head and induction head that somewhat compose. This", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 141, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "half-formed circuit becomes useful for reducing loss, so gradient descent completes the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 540, + 173, + 551 + ], + "spans": [ + { + "bbox": [ + 141, + 540, + 173, + 551 + ], + "score": 1.0, + "content": "circuit.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5, + "bbox_fs": [ + 133, + 506, + 505, + 551 + ] + }, + { + "type": "text", + "bbox": [ + 134, + 554, + 504, + 609 + ], + "lines": [ + { + "bbox": [ + 133, + 553, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 133, + 553, + 505, + 567 + ], + "score": 1.0, + "content": "• Evolution: A similar mystery arises from how organisms develop sophisticated machinery,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 141, + 564, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 141, + 564, + 505, + 578 + ], + "score": 1.0, + "content": "like the human eye. Each part is only useful in the context of other parts. A compelling", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 141, + 576, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 141, + 576, + 505, + 587 + ], + "score": 1.0, + "content": "explanation is a component first developed that was somewhat useful in its own right, like", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 141, + 586, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 141, + 586, + 505, + 600 + ], + "score": 1.0, + "content": "a light-detecting membrane. It was reinforced as a useful component. Then, later compo-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 141, + 598, + 396, + 611 + ], + "spans": [ + { + "bbox": [ + 141, + 598, + 396, + 611 + ], + "score": 1.0, + "content": "nents developed depending on the first, like the lens of the eye.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41, + "bbox_fs": [ + 133, + 553, + 505, + 611 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 618, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 617, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 631 + ], + "score": 1.0, + "content": "Evolution is a natural explanation, However, based on our toy tasks, it cannot be the whole story.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "In the repeated subsequence task, we have a sequence of uniform randomly generated tokens, apart", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "from a repeated subsequence at an arbitrary location, e.g. 7 2 8 3 1 9 3 8 3 1 9 9 2 5 END. This means", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "all pairs of tokens are independent, apart from pairs of equal tokens in the repeated subsequence. In", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 662, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 505, + 673 + ], + "score": 1.0, + "content": "particular, this means that a previous token head can never reduce loss for the current token. The", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "previous token will always be independent of the next token. So a previous token head is only useful", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 684, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 684, + 505, + 695 + ], + "score": 1.0, + "content": "in the context of an induction-like head that completes the circuit. Likewise, an induction head relies", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "on K-composition with a previous token head and so cannot be useful on its own. Yet the model", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 258, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 258, + 105 + ], + "score": 1.0, + "content": "eventually forms an induction circuit!", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 47, + "bbox_fs": [ + 105, + 617, + 506, + 695 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "on K-composition with a previous token head and so cannot be useful on its own. Yet the model", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 258, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 258, + 105 + ], + "score": 1.0, + "content": "eventually forms an induction circuit!", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 504, + 154 + ], + "lines": [ + { + "bbox": [ + 106, + 111, + 504, + 122 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 504, + 122 + ], + "score": 1.0, + "content": "A priori, the random walk seems insufficient on its own. An induction circuit is relatively compli-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 122, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 505, + 133 + ], + "score": 1.0, + "content": "cated, representing a small region in model space. So a random walk is unlikely to stumble upon", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "it. Concretely, in our modular addition case, progress measures show significant hidden progress", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 430, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 430, + 156 + ], + "score": 1.0, + "content": "pre-grokking, indicating the model did not stumble upon the solution by chance.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 160, + 505, + 215 + ], + "lines": [ + { + "bbox": [ + 105, + 159, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 173 + ], + "score": 1.0, + "content": "Thus, the lottery ticket hypothesis seems the most explanatory. An induction head is useless without", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 171, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 184 + ], + "score": 1.0, + "content": "a previous token head but might be slightly useful when composing with a head that uniformly", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 194 + ], + "score": 1.0, + "content": "attends to prior tokens, since part of its output will include the previous token! Nevertheless, we", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 193, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 505, + 205 + ], + "score": 1.0, + "content": "suspect that all explanations contribute to the entire picture. This seems most plausible if the uniform", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 204, + 436, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 436, + 217 + ], + "score": 1.0, + "content": "head just so happens to attend a bit more to the previous token via a random walk.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 221, + 505, + 287 + ], + "lines": [ + { + "bbox": [ + 106, + 222, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 505, + 233 + ], + "score": 1.0, + "content": "Returning to phase transitions, the lottery ticket-style explanation suggests that we might expect", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 232, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 505, + 244 + ], + "score": 1.0, + "content": "phase transitions as circuits form. Early in circuit formation, each part of the circuit is rough, so the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "effect on the loss of improving any individual component is weak, meaning gradients will be small.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 254, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 266 + ], + "score": 1.0, + "content": "As each component develops, other components will become more useful, meaning all gradients will", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 265, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 278 + ], + "score": 1.0, + "content": "increase together non-linearly. 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That being said, in cases where mechanistic interpretability does recover", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 141, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "human-understandable mechanisms, we could simply use the parts of the mechanism as", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 485, + 220, + 495 + ], + "spans": [ + { + "bbox": [ + 141, + 485, + 220, + 495 + ], + "score": 1.0, + "content": "progress measures.", + "type": "text" + } + ], + "index": 31, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 496, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 128, + 496, + 505, + 511 + ], + "score": 1.0, + "content": "2. If future mechanistic interpretability can only recover parts of the mechanism of larger", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 507, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 141, + 507, + 506, + 522 + ], + "score": 1.0, + "content": "models (as in Wang et al. (2022)) and can only generate comprehensive understanding", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 141, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 141, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "of the mechanisms of smaller models, we might still be able to use our understanding", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 141, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "from smaller models to guide the development measures that track parts of the behavior of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 141, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "the larger model. We find this scenario relatively plausible, as existing mechanistic inter-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 141, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "pretability work already allows us to recover fragments of large model behavior and un-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 141, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "derstand these fragments by analogy to smaller models. For example, Olsson et al. (2022)", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 141, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 141, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "use this approach to understand the emergence of in-context learning in medium-sized lan-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 141, + 586, + 225, + 597 + ], + "spans": [ + { + "bbox": [ + 141, + 586, + 225, + 597 + ], + "score": 1.0, + "content": "guage transformers.", + "type": "text" + } + ], + "index": 40, + "is_list_end_line": true + }, + { + "bbox": [ + 129, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 129, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "3. Even if mechanistic interpretability fails to recover understandable mechanisms at all on", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 141, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "large models, we might still be able to derive progress measures that don’t require human", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 141, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 141, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "understanding. For example, if we end up with automated mechanistic interpretability (that", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 141, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 141, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "nonetheless still fails to recover human-understandable mechanisms), we might be able to", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 141, + 644, + 313, + 656 + ], + "spans": [ + { + "bbox": [ + 141, + 644, + 313, + 656 + ], + "score": 1.0, + "content": "use the outputs of those opaque processes.", + "type": "text" + } + ], + "index": 45, + "is_list_end_line": true + }, + { + "bbox": [ + 141, + 656, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 141, + 656, + 505, + 670 + ], + "score": 1.0, + "content": "Another approach is task-independent progress measures: if we can discover progress mea-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 141, + 667, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 141, + 667, + 506, + 681 + ], + "score": 1.0, + "content": "sures that don’t depend on the task, perhaps using many small, interpretable models as", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 141, + 678, + 450, + 691 + ], + "spans": [ + { + "bbox": [ + 141, + 678, + 450, + 691 + ], + "score": 1.0, + "content": "testbeds, we might be able to apply these progress measures to large models.", + "type": "text" + } + ], + "index": 48, + "is_list_end_line": true + } + ], + "index": 36, + "bbox_fs": [ + 128, + 406, + 506, + 691 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 696, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 505, + 713 + ], + "score": 1.0, + "content": "That being said, we think the future work outlined in Section 6 is necessary to successfully apply", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "our approach to predict and understand emergent behavior in existing large language models, and so", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 721, + 225, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 225, + 733 + ], + "score": 1.0, + "content": "remain cautiously optimistic.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50, + "bbox_fs": [ + 105, + 696, + 505, + 733 + ] + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.2.2" +} \ No newline at end of file