Daily Papers

We study convergence lower bounds of without-replacement stochastic gradient descent (SGD) for solving smooth (strongly-)convex finite-sum minimization problems. Unlike most existing results focusing on final iterate lower bounds in terms of the number of components n and the number of epochs K, we seek bounds for arbitrary weighted average iterates that are tight in all factors including the condition number kappa. For SGD with Random Reshuffling, we present lower bounds that have tighter kappa dependencies than existing bounds. Our results are the first to perfectly close the gap between lower and upper bounds for weighted average iterates in both strongly-convex and convex cases. We also prove weighted average iterate lower bounds for arbitrary permutation-based SGD, which apply to all variants that carefully choose the best permutation. Our bounds improve the existing bounds in factors of n and kappa and thereby match the upper bounds shown for a recently proposed algorithm called GraB.

This work presents DAVINCI, a unified architecture for single-stage Computer-Aided Design (CAD) sketch parameterization and constraint inference directly from raster sketch images. By jointly learning both outputs, DAVINCI minimizes error accumulation and enhances the performance of constrained CAD sketch inference. Notably, DAVINCI achieves state-of-the-art results on the large-scale SketchGraphs dataset, demonstrating effectiveness on both precise and hand-drawn raster CAD sketches. To reduce DAVINCI's reliance on large-scale annotated datasets, we explore the efficacy of CAD sketch augmentations. We introduce Constraint-Preserving Transformations (CPTs), i.e. random permutations of the parametric primitives of a CAD sketch that preserve its constraints. This data augmentation strategy allows DAVINCI to achieve reasonable performance when trained with only 0.1% of the SketchGraphs dataset. Furthermore, this work contributes a new version of SketchGraphs, augmented with CPTs. The newly introduced CPTSketchGraphs dataset includes 80 million CPT-augmented sketches, thus providing a rich resource for future research in the CAD sketch domain.

The whole-brain connectome of a fruit fly comprises over 130K neurons connected with a probability of merely 0.02%, yet achieves an average shortest path of only 4.4 hops. Despite being highly structured at the circuit level, the network's long-range connections are broadly distributed across brain regions, functioning as stochastic shortcuts that enable efficient global communication. Inspired by this observation, we propose Stochastic Attention (SA), a drop-in enhancement for sliding-window attention (SWA) that applies a random permutation to the token sequence before windowed attention and restores the original order afterward. This transforms the fixed local window into a stochastic global one within the same O(nw) per-layer budget. Through depth, independently sampled permutations yield exponentially growing receptive fields, achieving full sequence coverage in O(log_w n) layers versus O(n/w) for SWA. We validate SA in two settings: pre-training language models from scratch, where a gated SA + SWA combination achieves the best average zero-shot accuracy, and training-free inference on Qwen3-8B and Qwen3-30B-A3B, where SA consistently outperforms SWA and matches or exceeds Mixture of Block Attention at comparable compute budgets. These results suggest that connectome-inspired stochastic routing is a practical primitive for improving the expressivity of efficient attention, complementary to existing linear and sparse approaches.

The output structure of database-like tables, consisting of values structured in horizontal rows and vertical columns identifiable by name, can cover a wide range of NLP tasks. Following this constatation, we propose a framework for text-to-table neural models applicable to problems such as extraction of line items, joint entity and relation extraction, or knowledge base population. The permutation-based decoder of our proposal is a generalized sequential method that comprehends information from all cells in the table. The training maximizes the expected log-likelihood for a table's content across all random permutations of the factorization order. During the content inference, we exploit the model's ability to generate cells in any order by searching over possible orderings to maximize the model's confidence and avoid substantial error accumulation, which other sequential models are prone to. Experiments demonstrate a high practical value of the framework, which establishes state-of-the-art results on several challenging datasets, outperforming previous solutions by up to 15%.

Recent studies have revealed the vulnerability of Deep Neural Networks (DNNs) to adversarial examples, which can easily fool DNNs into making incorrect predictions. To mitigate this deficiency, we propose a novel adversarial defense method called "Immunity" (Innovative MoE with MUtual information \& positioN stabilITY) based on a modified Mixture-of-Experts (MoE) architecture in this work. The key enhancements to the standard MoE are two-fold: 1) integrating of Random Switch Gates (RSGs) to obtain diverse network structures via random permutation of RSG parameters at evaluation time, despite of RSGs being determined after one-time training; 2) devising innovative Mutual Information (MI)-based and Position Stability-based loss functions by capitalizing on Grad-CAM's explanatory power to increase the diversity and the causality of expert networks. Notably, our MI-based loss operates directly on the heatmaps, thereby inducing subtler negative impacts on the classification performance when compared to other losses of the same type, theoretically. Extensive evaluation validates the efficacy of the proposed approach in improving adversarial robustness against a wide range of attacks.

In continual learning, catastrophic forgetting is affected by multiple aspects of the tasks. Previous works have analyzed separately how forgetting is affected by either task similarity or overparameterization. In contrast, our paper examines how task similarity and overparameterization jointly affect forgetting in an analyzable model. Specifically, we focus on two-task continual linear regression, where the second task is a random orthogonal transformation of an arbitrary first task (an abstraction of random permutation tasks). We derive an exact analytical expression for the expected forgetting - and uncover a nuanced pattern. In highly overparameterized models, intermediate task similarity causes the most forgetting. However, near the interpolation threshold, forgetting decreases monotonically with the expected task similarity. We validate our findings with linear regression on synthetic data, and with neural networks on established permutation task benchmarks.

Multimodal large language models (MLLMs) are now routinely deployed for visual understanding, generation, and curation. A substantial fraction of these applications require an explicit aesthetic judgment. Most existing solutions reduce this judgment to predicting a scalar score for a single image. We first ask whether such scores faithfully capture comparative preference: in a controlled study with eight expert annotators, score-derived rankings align poorly with the same annotators' direct comparisons, while direct ranking yields substantially higher inter-annotator agreement on best- and worst-image labels. Motivated by this finding, we introduce the Visual Aesthetic Benchmark (VAB), which casts aesthetic evaluation as comparative selection over candidate sets with matched subject matter. VAB contains 400 tasks and 1,195 images across fine art, photography, and illustration, with labels derived from the consensus of 10 independent expert judges per task. Evaluating 20 frontier MLLMs and six dedicated visual-quality reward models, we find that the strongest system identifies both the best and the worst image correctly across three random permutations of the candidate order in only 26.5% of tasks, far below the 68.9% achieved by human experts. Fine-tuning a 35B-parameter model on 2,000 expert examples brings its accuracy close to that of a 397B-parameter open-weight model, suggesting that the comparative signal in VAB is transferable. Together, these results expose a clear and measurable gap between current multimodal models and expert aesthetic judgment, and VAB provides the first set-based, expert-grounded testbed on which that gap can be tracked and closed.

Low-discrepancy (LD) sequences have been extensively used as efficient experimental designs across many scientific disciplines. QMCPy (https://qmcsoftware.github.io/QMCSoftware/) is an accessible Python library which provides a unified implementation of randomized LD sequences, automatic variable transformations, adaptive Quasi-Monte Carlo error estimation algorithms, and fast kernel methods. This article focuses on recent updates to QMCPy which broaden support for randomized LD sequences and add new tools to enable fast kernel methods using LD sequences. Specifically, we give a unified description of the supported LD lattices, digital nets, and Halton point sets, along with randomization options including random permutations / shifts, linear matrix scrambling (LMS), and nested uniform scrambling (NUS). We also support higher-order digital nets, higher-order scrambling with LMS or NUS, and Halton scrambling with LMS or NUS. For fast kernel methods, we provide shift-invariant (SI) and digitally-shift-invariant (DSI) kernels, including a new set of higher-order smoothness DSI kernels. When SI and DSI kernels are respectively paired with n LD lattice and digital net points, the resulting Gram matrices permit multiplication and inversion at only O(n log n) cost. These fast operations utilize QMCPy's implementation of the fast Fourier transform in bit-reversed order (FFTBR), inverse FFTBR (IFFTBR), and fast Walsh--Hadamard transform (FWHT).

Neural field is an emerging paradigm in data representation that trains a neural network to approximate the given signal. A key obstacle that prevents its widespread adoption is the encoding speed-generating neural fields requires an overfitting of a neural network, which can take a significant number of SGD steps to reach the desired fidelity level. In this paper, we delve into the impacts of data transformations on the speed of neural field training, specifically focusing on how permuting pixel locations affect the convergence speed of SGD. Counterintuitively, we find that randomly permuting the pixel locations can considerably accelerate the training. To explain this phenomenon, we examine the neural field training through the lens of PSNR curves, loss landscapes, and error patterns. Our analyses suggest that the random pixel permutations remove the easy-to-fit patterns, which facilitate easy optimization in the early stage but hinder capturing fine details of the signal.

Recent Vision Mamba models not only have much lower complexity for processing higher resolution images and longer videos but also the competitive performance with Vision Transformers (ViTs). However, they are stuck into overfitting and thus only present up to base size (about 80M). It is still unclear how vanilla Vision Mamba (Vim) can be efficiently scaled up to larger sizes, which is essentially for further exploitation. In this paper, we propose a stochastic layer-wise shuffle regularization, which empowers successfully scaling non-hierarchical Vision Mamba to a large size (about 300M) in a supervised setting. Specifically, our base and large-scale ShuffleMamba models can outperform the supervised ViTs of similar size by 0.8\% and 1.0\% classification accuracy on ImageNet1k, respectively, without auxiliary data. When evaluated on the ADE20K semantic segmentation and COCO detection tasks, our ShuffleMamba models also show significant improvements. Without bells and whistles, the stochastic layer-wise shuffle has the following highlights: (1) Plug and play: it does not change model architectures and will be omitted in inference. (2) Simple but effective: it can improve the overfitting in Vim training and only introduce random token permutation operations. (3) Intuitive: the token sequences in deeper layers are more likely to be shuffled as they are expected to be more semantic and less sensitive to patch positions. Code and models will be available at https://github.com/huangzizheng01/ShuffleMamba.

Large-scale models are pretrained on massive web-crawled datasets containing documents of mixed quality, making data filtering essential. A popular method is Classifier-based Quality Filtering (CQF), which trains a binary classifier to distinguish between pretraining data and a small, high-quality set. It assigns each pretraining document a quality score defined as the classifier's score and retains only the top-scoring ones. We provide an in-depth analysis of CQF. We show that while CQF improves downstream task performance, it does not necessarily enhance language modeling on the high-quality dataset. We explain this paradox by the fact that CQF implicitly filters the high-quality dataset as well. We further compare the behavior of models trained with CQF to those trained on synthetic data of increasing quality, obtained via random token permutations, and find starkly different trends. Our results challenge the view that CQF captures a meaningful notion of data quality.

Recent developments in Language Models (LMs) have shown their effectiveness in NLP tasks, particularly in knowledge-intensive tasks. However, the mechanisms underlying knowledge storage and memory access within their parameters remain elusive. In this paper, we investigate whether a generative LM (e.g., GPT-2) is able to access its memory sequentially or randomly. Through carefully-designed synthetic tasks, covering the scenarios of full recitation, selective recitation and grounded question answering, we reveal that LMs manage to sequentially access their memory while encountering challenges in randomly accessing memorized content. We find that techniques including recitation and permutation improve the random memory access capability of LMs. Furthermore, by applying this intervention to realistic scenarios of open-domain question answering, we validate that enhancing random access by recitation leads to notable improvements in question answering. The code to reproduce our experiments can be found at https://github.com/sail-sg/lm-random-memory-access.

Large language models (LLMs) exhibit positional bias in how they use context, which especially complicates listwise ranking. To address this, we propose permutation self-consistency, a form of self-consistency over ranking list outputs of black-box LLMs. Our key idea is to marginalize out different list orders in the prompt to produce an order-independent ranking with less positional bias. First, given some input prompt, we repeatedly shuffle the list in the prompt and pass it through the LLM while holding the instructions the same. Next, we aggregate the resulting sample of rankings by computing the central ranking closest in distance to all of them, marginalizing out prompt order biases in the process. Theoretically, we prove the robustness of our method, showing convergence to the true ranking in the presence of random perturbations. Empirically, on five list-ranking datasets in sorting and passage reranking, our approach improves scores from conventional inference by up to 7-18% for GPT-3.5 and 8-16% for LLaMA v2 (70B), surpassing the previous state of the art in passage reranking. Our code is at https://github.com/castorini/perm-sc.

Attention layers -- which map a sequence of inputs to a sequence of outputs -- are core building blocks of the Transformer architecture which has achieved significant breakthroughs in modern artificial intelligence. This paper presents a rigorous theoretical study on the learning and generalization of a single multi-head attention layer, with a sequence of key vectors and a separate query vector as input. We consider the random feature setting where the attention layer has a large number of heads, with randomly sampled frozen query and key matrices, and trainable value matrices. We show that such a random-feature attention layer can express a broad class of target functions that are permutation invariant to the key vectors. We further provide quantitative excess risk bounds for learning these target functions from finite samples, using random feature attention with finitely many heads. Our results feature several implications unique to the attention structure compared with existing random features theory for neural networks, such as (1) Advantages in the sample complexity over standard two-layer random-feature networks; (2) Concrete and natural classes of functions that can be learned efficiently by a random-feature attention layer; and (3) The effect of the sampling distribution of the query-key weight matrix (the product of the query and key matrix), where Gaussian random weights with a non-zero mean result in better sample complexities over the zero-mean counterpart for learning certain natural target functions. Experiments on simulated data corroborate our theoretical findings and further illustrate the interplay between the sample size and the complexity of the target function.

A key to knowledge graph embedding (KGE) is to choose a proper representation space, e.g., point-wise Euclidean space and complex vector space. In this paper, we propose a unified perspective of embedding and introduce uncertainty into KGE from the view of group theory. Our model can incorporate existing models (i.e., generality), ensure the computation is tractable (i.e., efficiency) and enjoy the expressive power of complex random variables (i.e., expressiveness). The core idea is that we embed entities/relations as elements of a symmetric group, i.e., permutations of a set. Permutations of different sets can reflect different properties of embedding. And the group operation of symmetric groups is easy to compute. In specific, we show that the embedding of many existing models, point vectors, can be seen as elements of a symmetric group. To reflect uncertainty, we first embed entities/relations as permutations of a set of random variables. A permutation can transform a simple random variable into a complex random variable for greater expressiveness, called a normalizing flow. We then define scoring functions by measuring the similarity of two normalizing flows, namely NFE. We construct several instantiating models and prove that they are able to learn logical rules. Experimental results demonstrate the effectiveness of introducing uncertainty and our model. The code is available at https://github.com/changyi7231/NFE.

Integrating heterogeneous biomedical data including imaging, omics, and clinical records supports accurate diagnosis and personalised care. Graph-based models fuse such non-Euclidean data by capturing spatial and relational structure, yet clinical uptake requires regulator-ready interpretability. We present the first technical survey of interpretable graph based models for multimodal biomedical data, covering 26 studies published between Jan 2019 and Sep 2024. Most target disease classification, notably cancer and rely on static graphs from simple similarity measures, while graph-native explainers are rare; post-hoc methods adapted from non-graph domains such as gradient saliency, and SHAP predominate. We group existing approaches into four interpretability families, outline trends such as graph-in-graph hierarchies, knowledge-graph edges, and dynamic topology learning, and perform a practical benchmark. Using an Alzheimer disease cohort, we compare Sensitivity Analysis, Gradient Saliency, SHAP and Graph Masking. SHAP and Sensitivity Analysis recover the broadest set of known AD pathways and Gene-Ontology terms, whereas Gradient Saliency and Graph Masking surface complementary metabolic and transport signatures. Permutation tests show all four beat random gene sets, but with distinct trade-offs: SHAP and Graph Masking offer deeper biology at higher compute cost, while Gradient Saliency and Sensitivity Analysis are quicker though coarser. We also provide a step-by-step flowchart covering graph construction, explainer choice and resource budgeting to help researchers balance transparency and performance. This review synthesises the state of interpretable graph learning for multimodal medicine, benchmarks leading techniques, and charts future directions, from advanced XAI tools to under-studied diseases, serving as a concise reference for method developers and translational scientists.

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of M^n modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces. We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov-Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Fr\'echet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.

Distributed applications need identifiers that satisfy storage efficiency, chronological sortability, origin metadata embedding, zero-lookup verifiability, confidentiality for external consumers, and multi-century addressability. Based on our literature survey, no existing scheme provides all six of these identifier properties within a unified system. This paper introduces Source Known Identifiers (SKIDs), a three-tier identity system that projects a single entity identity across trust boundaries, addressing all six properties. The first tier, Source Known ID (SKID), is a 64-bit signed integer embedding a timestamp with a 250-millisecond precision, application topology, and a per-entity-type sequence counter. It serves as the database primary key, providing compact storage (8 bytes) and natural B-tree ordering for optimized database indexing. The second tier, Source Known Entity ID (SKEID), extends the SKID into a 128-bit Universally Unique Identifier (UUID) compatible value by adding an entity type discriminator, an epoch selector, and a BLAKE3 keyed message authentication code (MAC). SKEIDs enable zero-lookup verification of identifier origin, integrity, and entity type within trusted environments, with a big-endian byte layout that preserves chronological ordering in lexicographic UUID string comparisons. The third tier, Secure SKEID, encrypts the entire SKEID using AES-256 symmetric encryption as a single-block pseudorandom permutation, producing ciphertext indistinguishable from random bytes while remaining compatible with standard UUID data-type parsers in string representation. Deterministic bidirectional transformations connect all three tiers.

A new information theoretic condition is presented for reconstructing a discrete random variable X based on the knowledge of a set of discrete functions of X. The reconstruction condition is derived from Shannon's 1953 lattice theory with two entropic metrics of Shannon and Rajski. Because such a theoretical material is relatively unknown and appears quite dispersed in different references, we first provide a synthetic description (with complete proofs) of its concepts, such as total, common and complementary informations. Definitions and properties of the two entropic metrics are also fully detailed and shown compatible with the lattice structure. A new geometric interpretation of such a lattice structure is then investigated that leads to a necessary (and sometimes sufficient) condition for reconstructing the discrete random variable X given a set { X_1,ldots,X_{n} } of elements in the lattice generated by X. Finally, this condition is illustrated in five specific examples of perfect reconstruction problems: reconstruction of a symmetric random variable from the knowledge of its sign and absolute value, reconstruction of a word from a set of linear combinations, reconstruction of an integer from its prime signature (fundamental theorem of arithmetic) and from its remainders modulo a set of coprime integers (Chinese remainder theorem), and reconstruction of the sorting permutation of a list from a minimal set of pairwise comparisons.

The ability of an architecture to realize permutations is quite fundamental. For example, Large Language Models need to be able to correctly copy (and perhaps rearrange) parts of the input prompt into the output. Classical universal approximation theorems guarantee the existence of parameter configurations that solve this task but offer no insights into whether gradient-based algorithms can find them. In this paper, we address this gap by focusing on two-layer fully connected feed-forward neural networks and the task of learning permutations on nonzero binary inputs. We show that in the infinite width Neural Tangent Kernel (NTK) regime, an ensemble of such networks independently trained with gradient descent on only the k standard basis vectors out of 2^k - 1 possible inputs successfully learns any fixed permutation of length k with arbitrarily high probability. By analyzing the exact training dynamics, we prove that the network's output converges to a Gaussian process whose mean captures the ground truth permutation via sign-based features. We then demonstrate how averaging these runs (an "ensemble" method) and applying a simple rounding step yields an arbitrarily accurate prediction on any possible input unseen during training. Notably, the number of models needed to achieve exact learning with high probability (which we refer to as ensemble complexity) exhibits a linearithmic dependence on the input size k for a single test input and a quadratic dependence when considering all test inputs simultaneously.

Large language models perform near-Bayesian inference yet violate permutation invariance on exchangeable data. We resolve this by showing transformers minimize expected conditional description length (cross-entropy) over orderings, E_pi[ell(Y mid Gamma_pi(X))], which admits a Kolmogorov-complexity interpretation up to additive constants, rather than the permutation-invariant description length ell(Y mid X). This makes them Bayesian in expectation, not in realization. We derive (i) a Quantified Martingale Violation bound showing order-induced deviations scale as O(log n) with constants; (ii) the Expectation-level Decompression Law linking information budgets to reliability for Bernoulli predicates; and (iii) deployable planners (B2T/RoH/ISR) for answer/abstain decisions. Empirically, permutation dispersion follows a+bln n (Qwen2-7B b approx 0.377, Llama-3.1-8B b approx 0.147); permutation mixtures improve ground-truth likelihood/accuracy; and randomized dose-response shows hallucinations drop by sim 0.13 per additional nat. A pre-specified audit with a fixed ISR=1.0 achieves near-0\% hallucinations via calibrated refusal at 24\% abstention. The framework turns hallucinations into predictable compression failures and enables principled information budgeting.

In this article, we propose a new counter-based implementation of John von Neumann's middle-square random number generator (RNG). Several rounds of squaring are applied to a counter to produce a random output. We discovered that four rounds are sufficient to provide satisfactory data. Two versions of the RNG are presented, a 4-round version with 32-bit output and a 5-round version with 64-bit output. Both pass stringent tests of randomness and may be the fastest counter-based generators.

We study the problem of detecting the correlation between two Gaussian databases XinR^{ntimes d} and Y^{ntimes d}, each composed of n users with d features. This problem is relevant in the analysis of social media, computational biology, etc. We formulate this as a hypothesis testing problem: under the null hypothesis, these two databases are statistically independent. Under the alternative, however, there exists an unknown permutation sigma over the set of n users (or, row permutation), such that X is rho-correlated with Y^sigma, a permuted version of Y. We determine sharp thresholds at which optimal testing exhibits a phase transition, depending on the asymptotic regime of n and d. Specifically, we prove that if rho^2dto0, as dtoinfty, then weak detection (performing slightly better than random guessing) is statistically impossible, irrespectively of the value of n. This compliments the performance of a simple test that thresholds the sum all entries of X^TY. Furthermore, when d is fixed, we prove that strong detection (vanishing error probability) is impossible for any rho<rho^star, where rho^star is an explicit function of d, while weak detection is again impossible as long as rho^2dto0. These results close significant gaps in current recent related studies.

We introduce a framework for automatically defining and learning deep generative models with problem-specific structure. We tackle problem domains that are more traditionally solved by algorithms such as sorting, constraint satisfaction for Sudoku, and matrix factorization. Concretely, we train diffusion models with an architecture tailored to the problem specification. This problem specification should contain a graphical model describing relationships between variables, and often benefits from explicit representation of subcomputations. Permutation invariances can also be exploited. Across a diverse set of experiments we improve the scaling relationship between problem dimension and our model's performance, in terms of both training time and final accuracy. Our code can be found at https://github.com/plai-group/gsdm.

We present a principled approach to uncover the structure of visual data by solving a novel deep learning task coined visual permutation learning. The goal of this task is to find the permutation that recovers the structure of data from shuffled versions of it. In the case of natural images, this task boils down to recovering the original image from patches shuffled by an unknown permutation matrix. Unfortunately, permutation matrices are discrete, thereby posing difficulties for gradient-based methods. To this end, we resort to a continuous approximation of these matrices using doubly-stochastic matrices which we generate from standard CNN predictions using Sinkhorn iterations. Unrolling these iterations in a Sinkhorn network layer, we propose DeepPermNet, an end-to-end CNN model for this task. The utility of DeepPermNet is demonstrated on two challenging computer vision problems, namely, (i) relative attributes learning and (ii) self-supervised representation learning. Our results show state-of-the-art performance on the Public Figures and OSR benchmarks for (i) and on the classification and segmentation tasks on the PASCAL VOC dataset for (ii).

Randomized controlled trials are susceptible to imbalance on covariates predictive of the outcome. Rerandomization and deterministic treatment assignment are two proposed solutions. This paper explores the relationship between rerandomization and deterministic assignment, showing how deterministic assignment is an extreme case of rerandomization. The paper argues that in small experiments, both fully randomized and fully deterministic assignment have limitations. Instead, the researcher should consider setting the rerandomization acceptance probability based on an analysis of covariates and assumptions about the data structure to achieve an optimal alignment between randomness and balance. This allows for the calculation of minimum p-values along with valid permutation tests and fiducial intervals. The paper also introduces tools, including a new, open-source R package named fastrerandomize, to implement rerandomization and explore options for optimal rerandomization acceptance thresholds.

Diffusion models based on permutation-equivariant networks can learn permutation-invariant distributions for graph data. However, in comparison to their non-invariant counterparts, we have found that these invariant models encounter greater learning challenges since 1) their effective target distributions exhibit more modes; 2) their optimal one-step denoising scores are the score functions of Gaussian mixtures with more components. Motivated by this analysis, we propose a non-invariant diffusion model, called SwinGNN, which employs an efficient edge-to-edge 2-WL message passing network and utilizes shifted window based self-attention inspired by SwinTransformers. Further, through systematic ablations, we identify several critical training and sampling techniques that significantly improve the sample quality of graph generation. At last, we introduce a simple post-processing trick, i.e., randomly permuting the generated graphs, which provably converts any graph generative model to a permutation-invariant one. Extensive experiments on synthetic and real-world protein and molecule datasets show that our SwinGNN achieves state-of-the-art performances. Our code is released at https://github.com/qiyan98/SwinGNN.

We study the problem of designing models for machine learning tasks defined on sets. In contrast to traditional approach of operating on fixed dimensional vectors, we consider objective functions defined on sets that are invariant to permutations. Such problems are widespread, ranging from estimation of population statistics poczos13aistats, to anomaly detection in piezometer data of embankment dams Jung15Exploration, to cosmology Ntampaka16Dynamical,Ravanbakhsh16ICML1. Our main theorem characterizes the permutation invariant functions and provides a family of functions to which any permutation invariant objective function must belong. This family of functions has a special structure which enables us to design a deep network architecture that can operate on sets and which can be deployed on a variety of scenarios including both unsupervised and supervised learning tasks. We also derive the necessary and sufficient conditions for permutation equivariance in deep models. We demonstrate the applicability of our method on population statistic estimation, point cloud classification, set expansion, and outlier detection.

In this paper, we provide a finite random iterated function system satisfying the open set condition, for which the random version of Bowen's formula fails to hold. This counterexample shows that analogous results established for random recursive constructions are not always obtained for random iterated function systems.

Randomized mechanisms can have good normative properties compared to their deterministic counterparts. However, randomized mechanisms are problematic in several ways such as in their verifiability. We propose here to derandomize such mechanisms by having agents play a game instead of tossing a coin. The game is designed so an agent's best action is to play randomly, and this play then injects ``randomness'' into the mechanism. This derandomization retains many of the good normative properties of the original randomized mechanism but gives a mechanism that is deterministic and easy, for instance, to audit. We consider three related methods to derandomize randomized mechanism in six different domains: voting, facility location, task allocation, school choice, peer selection, and resource allocation. We propose a number of novel derandomized mechanisms for these six domains with good normative properties. Each mechanism has a mixed Nash equilibrium in which agents play a modular arithmetic game with an uniform mixed strategy. In all but one mixed Nash equilibrium, agents report their preferences over the original problem sincerely. The derandomized methods are thus ``quasi-strategy proof''. In one domain, we additionally show that a new and desirable normative property emerges as a result of derandomization.

Generating functions, which are widely used in combinatorics and probability theory, encode function values into the coefficients of a polynomial. In this paper, we explore their use as a tractable probabilistic model, and propose probabilistic generating circuits (PGCs) for their efficient representation. PGCs are strictly more expressive efficient than many existing tractable probabilistic models, including determinantal point processes (DPPs), probabilistic circuits (PCs) such as sum-product networks, and tractable graphical models. We contend that PGCs are not just a theoretical framework that unifies vastly different existing models, but also show great potential in modeling realistic data. We exhibit a simple class of PGCs that are not trivially subsumed by simple combinations of PCs and DPPs, and obtain competitive performance on a suite of density estimation benchmarks. We also highlight PGCs' connection to the theory of strongly Rayleigh distributions.

We propose Monte Carlo Permutation Search (MCPS), a general-purpose Monte Carlo Tree Search (MCTS) algorithm that improves upon the GRAVE algorithm. MCPS is relevant when deep reinforcement learning is not an option, or when the computing power available before play is not substantial, such as in General Game Playing, for example. The principle of MCPS is to include in the exploration term of a node the statistics on all the playouts that contain all the moves on the path from the root to the node. We extensively test MCPS on a variety of games: board games, wargame, investment game, video game and multi-player games. MCPS has better results than GRAVE in all the two-player games. It has equivalent results for multi-player games because these games are inherently balanced even when players have different strengths. We also show that using abstract codes for moves instead of exact codes can be beneficial to both MCPS and GRAVE, as they improve the permutation statistics and the AMAF statistics. We also provide a mathematical derivation of the formulas used for weighting the three sources of statistics. These formulas are an improvement on the GRAVE formula since they no longer use the bias hyperparameter of GRAVE. Moreover, MCPS is not sensitive to the ref hyperparameter.

Proportionality is an attractive fairness concept that has been applied to a range of problems including the facility location problem, a classic problem in social choice. In our work, we propose a concept called Strong Proportionality, which ensures that when there are two groups of agents at different locations, both groups incur the same total cost. We show that although Strong Proportionality is a well-motivated and basic axiom, there is no deterministic strategyproof mechanism satisfying the property. We then identify a randomized mechanism called Random Rank (which uniformly selects a number k between 1 to n and locates the facility at the k'th highest agent location) which satisfies Strong Proportionality in expectation. Our main theorem characterizes Random Rank as the unique mechanism that achieves universal truthfulness, universal anonymity, and Strong Proportionality in expectation among all randomized mechanisms. Finally, we show via the AverageOrRandomRank mechanism that even stronger ex-post fairness guarantees can be achieved by weakening universal truthfulness to strategyproofness in expectation.

Modern language models define distributions over strings, but downstream tasks often require different output formats. For instance, a model that generates byte-pair strings does not directly produce word-level predictions, and a DNA model does not directly produce amino-acid sequences. In such cases, a deterministic string-to-string transformation can convert the model's output to the desired form. This is a familiar pattern in probability theory: applying a function f to a random variable Xsim p yields a transformed random variable f(X) with an induced distribution. While such transformations are occasionally used in language modeling, prior work does not treat them as yielding new, fully functional language models. We formalize this perspective and introduce a general framework for language models derived from deterministic string-to-string transformations. We focus on transformations representable as finite-state transducers -- a commonly used state-machine abstraction for efficient string-to-string mappings. We develop algorithms that compose a language model with an FST to *marginalize* over source strings mapping to a given target, propagating probabilities through the transducer without altering model parameters and enabling *conditioning* on transformed outputs. We present an exact algorithm, an efficient approximation, and a theoretical analysis. We conduct experiments in three domains: converting language models from tokens to bytes, from tokens to words, and from DNA to amino acids. These experiments demonstrate inference-time adaptation of pretrained language models to match application-specific output requirements.

In complex systems, we often observe complex global behavior emerge from a collection of agents interacting with each other in their environment, with each individual agent acting only on locally available information, without knowing the full picture. Such systems have inspired development of artificial intelligence algorithms in areas such as swarm optimization and cellular automata. Motivated by the emergence of collective behavior from complex cellular systems, we build systems that feed each sensory input from the environment into distinct, but identical neural networks, each with no fixed relationship with one another. We show that these sensory networks can be trained to integrate information received locally, and through communication via an attention mechanism, can collectively produce a globally coherent policy. Moreover, the system can still perform its task even if the ordering of its inputs is randomly permuted several times during an episode. These permutation invariant systems also display useful robustness and generalization properties that are broadly applicable. Interactive demo and videos of our results: https://attentionneuron.github.io/

We introduce the concurrent shuffle model of differential privacy. In this model we have multiple concurrent shufflers permuting messages from different, possibly overlapping, batches of users. Similarly to the standard (single) shuffle model, the privacy requirement is that the concatenation of all shuffled messages should be differentially private. We study the private continual summation problem (a.k.a. the counter problem) and show that the concurrent shuffle model allows for significantly improved error compared to a standard (single) shuffle model. Specifically, we give a summation algorithm with error O(n^{1/(2k+1)}) with k concurrent shufflers on a sequence of length n. Furthermore, we prove that this bound is tight for any k, even if the algorithm can choose the sizes of the batches adaptively. For k=log n shufflers, the resulting error is polylogarithmic, much better than Theta(n^{1/3}) which we show is the smallest possible with a single shuffler. We use our online summation algorithm to get algorithms with improved regret bounds for the contextual linear bandit problem. In particular we get optimal O(n) regret with k= Omega(log n) concurrent shufflers.

Citizens' assemblies are a form of democratic innovation in which a randomly selected panel of constituents deliberates on questions of public interest. We study a novel goal for the selection of panel members: maximizing the entropy of the distribution over possible panels. We design algorithms that sample from maximum-entropy distributions, potentially subject to constraints on the individual selection probabilities. We investigate the properties of these algorithms theoretically, including in terms of their resistance to manipulation and transparency. We benchmark our algorithms on a large set of real assembly lotteries in terms of their intersectional diversity and the probability of satisfying unseen representation constraints, and we obtain favorable results on both measures. We deploy one of our algorithms on a website for citizens' assembly practitioners.

Self-supervised learning of convolutional neural networks can harness large amounts of cheap unlabeled data to train powerful feature representations. As surrogate task, we jointly address ordering of visual data in the spatial and temporal domain. The permutations of training samples, which are at the core of self-supervision by ordering, have so far been sampled randomly from a fixed preselected set. Based on deep reinforcement learning we propose a sampling policy that adapts to the state of the network, which is being trained. Therefore, new permutations are sampled according to their expected utility for updating the convolutional feature representation. Experimental evaluation on unsupervised and transfer learning tasks demonstrates competitive performance on standard benchmarks for image and video classification and nearest neighbor retrieval.

In this work, a method of random parameters generation for randomized learning of a single-hidden-layer feedforward neural network is proposed. The method firstly, randomly selects the slope angles of the hidden neurons activation functions from an interval adjusted to the target function, then randomly rotates the activation functions, and finally distributes them across the input space. For complex target functions the proposed method gives better results than the approach commonly used in practice, where the random parameters are selected from the fixed interval. This is because it introduces the steepest fragments of the activation functions into the input hypercube, avoiding their saturation fragments.

For a sequence of random variables (X_1, X_2, ldots, X_n), n geq 1, that are independent and identically distributed with a regularly varying tail with index -alpha, alpha geq 0, we show that the contribution of the maximum term M_n triangleq max(X_1,ldots,X_n) in the sum S_n triangleq X_1 + cdots +X_n, as n to infty, decreases monotonically with alpha in stochastic ordering sense.

The Omega numbers-the halting probabilities of universal prefix-free machines-are known to be exactly the Martin-L{\"o}f random left-c.e. reals. We show that one cannot uniformly produce, from a Martin-L{\"o}f random left-c.e. real alpha, a universal prefix-free machine U whose halting probability is alpha. We also answer a question of Barmpalias and Lewis-Pye by showing that given a left-c.e. real alpha, one cannot uniformly produce a left-c.e. real beta such that alpha -- beta is neither left-c.e. nor right-c.e.

The aim of this paper is to present an elementary computable theory of probability, random variables and stochastic processes. The probability theory is baed on existing approaches using valuations and lower integrals. Various approaches to random variables are discussed, including the approach based on completions in a Polish space. We apply the theory to the study of stochastic dynamical systems in discrete-time, and give a brief exposition of the Wiener process as a foundation for stochastic differential equations. The theory is based within the framework of type-two effectivity, so has an explicit direct link with Turing computation, and is expressed in a system of computable types and operations, so has a clean mathematical description.

In this technical note, we study the problem of inverse permutation learning in decoder-only transformers. Given a permutation and a string to which that permutation has been applied, the model is tasked with producing the original (``canonical'') string. We argue that this task models a natural robustness property across a variety of reasoning tasks, including long-context retrieval, multiple choice QA and in-context learning. Our primary contribution is an impossibility result: we show that an arbitrary depth, decoder-only transformer cannot learn this task. This result concerns the expressive capacity of decoder-only transformer models and is agnostic to training dynamics or sample complexity. We give a pair of alternative constructions under which inverse permutation learning is feasible. The first of these highlights the fundamental role of the causal attention mask, and reveals a gap between the expressivity of encoder-decoder transformers and the more popular decoder-only architecture. The latter result is more surprising: we show that simply padding the input with ``scratch tokens" yields a construction under which inverse permutation learning is possible. We conjecture that this may suggest an alternative mechanism by which chain-of-thought prompting or, more generally, intermediate ``thinking'' tokens can enable reasoning in large language models, even when these tokens encode no meaningful semantic information (e.g., the results of intermediate computations).

We revisit the noisy binary search model of Karp and Kleinberg, in which we have n coins with unknown probabilities p_i that we can flip. The coins are sorted by increasing p_i, and we would like to find where the probability crosses (to within varepsilon) of a target value tau. This generalized the fixed-noise model of Burnashev and Zigangirov , in which p_i = 1{2} pm varepsilon, to a setting where coins near the target may be indistinguishable from it. Karp and Kleinberg showed that Theta(1{varepsilon^2} log n) samples are necessary and sufficient for this task. We produce a practical algorithm by solving two theoretical challenges: high-probability behavior and sharp constants. We give an algorithm that succeeds with probability 1-delta from \[ 1{C_{\tau, \varepsilon}} \cdot \left(\lg n + O(\log^{2/3} n \log^{1/3} 1{\delta} + \log 1{\delta})\right) \] samples, where C_{tau, varepsilon} is the optimal such constant achievable. For delta > n^{-o(1)} this is within 1 + o(1) of optimal, and for delta ll 1 it is the first bound within constant factors of optimal.

Language model training in distributed settings is limited by the communication cost of gradient exchanges. In this short note, we extend recent work from Malladi et al. (2023), using shared randomness to perform distributed fine-tuning with low bandwidth. The method is a natural decentralized extension of memory-efficient Simultaneous Perturbation Stochastic Approximation (SPSA). Each iteration, each machine seeds a Random Number Generator (RNG) to perform local reproducible perturbations on model weights and calculate and exchange scalar projected gradients, which are then used to update each model. By using a (machine, sample) identifier as the random seed, each model can regenerate one another's perturbations. As machines only exchange single-byte projected gradients, this is highly communication efficient. There are also potential privacy benefits, as projected gradients may be calculated on different training data, and models never access the other's data. Our approach not only drastically reduces communication bandwidth requirements but also accommodates dynamic addition or removal of machines during the training process and retains the memory-efficient and inference-only advantages of recent work. We perform proof-of-concept experiments to demonstrate the potential usefulness of this method, building off of rich literature on distributed optimization and memory-efficient training.

We study the set of networks, which consist of sources, sinks and neutral points, bijective to the permutations. The set of directed edges, which characterizes a network, is constructed from a polyomino or a Rothe diagram of a permutation through a Dyck tiling on a ribbon. We introduce a new combinatorial object similar to a tree-like tableau, which we call a forest. A forest is shown to give a permutation, and be bijective to a network corresponding to the inverse of the permutation. We show that the poset of networks is a finite graded lattice and admits an EL-labeling. By use of this EL-labeling, we show the lattice is supersolvable and compute the M\"obius function of an interval of the poset.

We study the classic Text-to-Pattern Hamming Distances problem: given a pattern P of length m and a text T of length n, both over a polynomial-size alphabet, compute the Hamming distance between P and T[i, ., . , i+m-1] for every shift i, under the standard Word-RAM model with Theta(log n)-bit words. - We provide an O(nm) time Las Vegas randomized algorithm for this problem, beating the decades-old O(n m log m) running time [Abrahamson, SICOMP 1987]. We also obtain a deterministic algorithm, with a slightly higher O(nm(log mloglog m)^{1/4}) running time. Our randomized algorithm extends to the k-bounded setting, with running time Obig(n+nk{m}big), removing all the extra logarithmic factors from earlier algorithms [Gawrychowski and Uzna\'{n}ski, ICALP 2018; Chan, Golan, Kociumaka, Kopelowitz and Porat, STOC 2020]. - For the (1+epsilon)-approximate version of Text-to-Pattern Hamming Distances, we give an O(epsilon^{-0.93}n) time Monte Carlo randomized algorithm, beating the previous O(epsilon^{-1}n) running time [Kopelowitz and Porat, FOCS 2015; Kopelowitz and Porat, SOSA 2018]. Our approximation algorithm exploits a connection with 3SUM, and uses a combination of Fredman's trick, equality matrix product, and random sampling; in particular, we obtain new results on approximate counting versions of 3SUM and Exact Triangle, which may be of independent interest. Our exact algorithms use a novel combination of hashing, bit-packed FFT, and recursion; in particular, we obtain a faster algorithm for computing the sumset of two integer sets, in the regime when the universe size is close to quadratic in the number of elements. We also prove a fine-grained equivalence between the exact Text-to-Pattern Hamming Distances problem and a range-restricted, counting version of 3SUM.

We analyze the DP (differential privacy) properties of the quantum recommendation algorithm and the quantum-inspired-classical recommendation algorithm. We discover that the quantum recommendation algorithm is a privacy curating mechanism on its own, requiring no external noise, which is different from traditional differential privacy mechanisms. In our analysis, a novel perturbation method tailored for SVD (singular value decomposition) and low-rank matrix approximation problems is introduced. Using the perturbation method and random matrix theory, we are able to derive that both the quantum and quantum-inspired-classical algorithms are big(mathcal{O}big(frac 1nbig),,, mathcal{O}big(1{min{m,n}}big)big)-DP under some reasonable restrictions, where m and n are numbers of users and products in the input preference database respectively. Nevertheless, a comparison shows that the quantum algorithm has better privacy preserving potential than the classical one.

The game show Lucky 13 differs from other television game shows in that contestants are required to place a bet on their own knowledge of trivia by selecting a range that contains the number of questions that they answered correctly. We present a model for this game show using binomial random variables and generate tables outlining the optimal range the player should select based on maximization of two different utility functions. After analyzing the decisions made by some actual contestants on this show, we present a numerical simulation for how many questions an average player is expected to answer correctly based on question categories observed for two sample contestants.

Finite symmetric groups S_n are essential in fields such as combinatorics, physics, and chemistry. However, learning a probability distribution over S_n poses significant challenges due to its intractable size and discrete nature. In this paper, we introduce SymmetricDiffusers, a novel discrete diffusion model that simplifies the task of learning a complicated distribution over S_n by decomposing it into learning simpler transitions of the reverse diffusion using deep neural networks. We identify the riffle shuffle as an effective forward transition and provide empirical guidelines for selecting the diffusion length based on the theory of random walks on finite groups. Additionally, we propose a generalized Plackett-Luce (PL) distribution for the reverse transition, which is provably more expressive than the PL distribution. We further introduce a theoretically grounded "denoising schedule" to improve sampling and learning efficiency. Extensive experiments show that our model achieves state-of-the-art or comparable performances on solving tasks including sorting 4-digit MNIST images, jigsaw puzzles, and traveling salesman problems. Our code is released at https://github.com/DSL-Lab/SymmetricDiffusers.

It has recently been conjectured that neural network solution sets reachable via stochastic gradient descent (SGD) are convex, considering permutation invariances (Entezari et al., 2022). This means that a linear path can connect two independent solutions with low loss, given the weights of one of the models are appropriately permuted. However, current methods to test this theory often require very wide networks to succeed. In this work, we conjecture that more generally, the SGD solution set is a "star domain" that contains a "star model" that is linearly connected to all the other solutions via paths with low loss values, modulo permutations. We propose the Starlight algorithm that finds a star model of a given learning task. We validate our claim by showing that this star model is linearly connected with other independently found solutions. As an additional benefit of our study, we demonstrate better uncertainty estimates on the Bayesian Model Averaging over the obtained star domain. Further, we demonstrate star models as potential substitutes for model ensembles. Our code is available at https://github.com/aktsonthalia/starlight.

The assumption that response and predictor belong to the same statistical unit may be violated in practice. Unbiased estimation and recovery of true label ordering based on unlabeled data are challenging tasks and have attracted increasing attentions in the recent literature. In this paper, we present a relatively complete analysis of label permutation problem for the generalized linear model with multivariate responses. The theory is established under different scenarios, with knowledge of true parameters, with partial knowledge of underlying label permutation matrix and without any knowledge. Our results remove the stringent conditions required by the current literature and are further extended to the missing observation setting which has never been considered in the field of label permutation problem. On computational side, we propose two methods, "maximum likelihood estimation" algorithm and "two-step estimation" algorithm, to accommodate for different settings. When the proportion of permuted labels is moderate, both methods work effectively. Multiple numerical experiments are provided and corroborate our theoretical findings.

Conformal prediction is a widely used method to quantify the uncertainty of a classifier under the assumption of exchangeability (e.g., IID data). We generalize conformal prediction to the Hidden Markov Model (HMM) framework where the assumption of exchangeability is not valid. The key idea of the proposed method is to partition the non-exchangeable Markovian data from the HMM into exchangeable blocks by exploiting the de Finetti's Theorem for Markov Chains discovered by Diaconis and Freedman (1980). The permutations of the exchangeable blocks are viewed as randomizations of the observed Markovian data from the HMM. The proposed method provably retains all desirable theoretical guarantees offered by the classical conformal prediction framework in both exchangeable and Markovian settings. In particular, while the lack of exchangeability introduced by Markovian samples constitutes a violation of a crucial assumption for classical conformal prediction, the proposed method views it as an advantage that can be exploited to improve the performance further. Detailed numerical and empirical results that complement the theoretical conclusions are provided to illustrate the practical feasibility of the proposed method.

A universal word (u-word) for d-dimensional permutations of length n is a 2-dimensional word with d-1 rows, any size n window of which is order-isomorphic to exactly one permutation of length n, and all permutations of length n are covered. It is known that u-words (in fact, even u-cycles, a stronger claim) for d-dimensional permutations exist. In this paper, we use the idea of incomparable elements to prove that u-words of length (n!)^{d-1}+n-1-i(n-1), for dgeq 2 and $0leq ileq 2^{d-1}{n-1}left[(1+(n-1)!)^{d-1}-left(1+(n-1)!{2}right)^{d-1}right], for d-dimensional permutations of length n exist, which generalizes the respective result of Kitaev, Potapov and Vajnovszki for ``usual'' permutations (d=2$).

We define a new class of set functions that in addition to being monotone and subadditive, also admit a very limited form of submodularity defined over a permutation of the ground set. We refer to this permutation as a submodular order. This class of functions includes monotone submodular functions as a sub-family. To understand the importance of this structure in optimization problems we consider the problem of maximizing function value under various types of constraints. To demonstrate the modeling power of submodular order functions we show applications in two different settings. First, we apply our results to the extensively studied problem of assortment optimization. While the objectives in assortment optimization are known to be non-submodular (and non-monotone) even for simple choice models, we show that they are compatible with the notion of submodular order. Consequently, we obtain new and in some cases the first constant factor guarantee for constrained assortment optimization in fundamental choice models. As a second application of submodular order functions, we show an intriguing connection to the maximization of monotone submodular functions in the streaming model. We recover some best known guarantees for this problem as a corollary of our results.

We introduce Minary, a computational framework designed as a candidate for the first formally provable autopoietic primitive. Minary represents interacting probabilistic events as multi-dimensional vectors and combines them via linear superposition rather than multiplicative scalar operations, thereby preserving uncertainty and enabling constructive and destructive interference in the range [-1,1]. A fixed set of ``perspectives'' evaluates ``semantic dimensions'' according to hidden competencies, and their interactions drive two discrete-time stochastic processes. We model this system as an iterated random affine map and use the theory of iterated random functions to prove that it converges in distribution to a unique stationary law; we moreover obtain an explicit closed form for the limiting expectation in terms of row, column, and global averages of the competency matrix. We then derive exact formulas for the mean and variance of the normalized consensus conditioned on the activation of a given semantic dimension, revealing how consensus depends on competency structure rather than raw input signals. Finally, we argue that Minary is organizationally closed yet operationally open in the sense of Maturana and Varela, and we discuss implications for building self-maintaining, distributed, and parallelizable computational systems that house a uniquely subjective notion of identity.

We propose prioritized unit propagation with periodic resetting, which is a simple but surprisingly effective algorithm for solving random SAT instances that are meant to be hard. In particular, an evaluation on the Random Track of the 2017 and 2018 SAT competitions shows that a basic prototype of this simple idea already ranks at second place in both years. We share this observation in the hope that it helps the SAT community better understand the hardness of random instances used in competitions and inspire other interesting ideas on SAT solving.

A challenging problem in many modern machine learning tasks is to process weight-space features, i.e., to transform or extract information from the weights and gradients of a neural network. Recent works have developed promising weight-space models that are equivariant to the permutation symmetries of simple feedforward networks. However, they are not applicable to general architectures, since the permutation symmetries of a weight space can be complicated by recurrence or residual connections. This work proposes an algorithm that automatically constructs permutation equivariant models, which we refer to as universal neural functionals (UNFs), for any weight space. Among other applications, we demonstrate how UNFs can be substituted into existing learned optimizer designs, and find promising improvements over prior methods when optimizing small image classifiers and language models. Our results suggest that learned optimizers can benefit from considering the (symmetry) structure of the weight space they optimize. We open-source our library for constructing UNFs at https://github.com/AllanYangZhou/universal_neural_functional.

Recently, Ainsworth et al. showed that using weight matching (WM) to minimize the L_2 distance in a permutation search of model parameters effectively identifies permutations that satisfy linear mode connectivity (LMC), in which the loss along a linear path between two independently trained models with different seeds remains nearly constant. This paper provides a theoretical analysis of LMC using WM, which is crucial for understanding stochastic gradient descent's effectiveness and its application in areas like model merging. We first experimentally and theoretically show that permutations found by WM do not significantly reduce the L_2 distance between two models and the occurrence of LMC is not merely due to distance reduction by WM in itself. We then provide theoretical insights showing that permutations can change the directions of the singular vectors, but not the singular values, of the weight matrices in each layer. This finding shows that permutations found by WM mainly align the directions of singular vectors associated with large singular values across models. This alignment brings the singular vectors with large singular values, which determine the model functionality, closer between pre-merged and post-merged models, so that the post-merged model retains functionality similar to the pre-merged models, making it easy to satisfy LMC. Finally, we analyze the difference between WM and straight-through estimator (STE), a dataset-dependent permutation search method, and show that WM outperforms STE, especially when merging three or more models.

Consistently checking the statistical significance of experimental results is one of the mandatory methodological steps to address the so-called "reproducibility crisis" in deep reinforcement learning. In this tutorial paper, we explain how the number of random seeds relates to the probabilities of statistical errors. For both the t-test and the bootstrap confidence interval test, we recall theoretical guidelines to determine the number of random seeds one should use to provide a statistically significant comparison of the performance of two algorithms. Finally, we discuss the influence of deviations from the assumptions usually made by statistical tests. We show that they can lead to inaccurate evaluations of statistical errors and provide guidelines to counter these negative effects. We make our code available to perform the tests.

Recent works show that ordering of the training data affects the model performance for Neural Machine Translation. Several approaches involving dynamic data ordering and data sharding based on curriculum learning have been analysed for the their performance gains and faster convergence. In this work we propose to empirically study several ordering approaches for the training data based on different metrics and evaluate their impact on the model performance. Results from our study show that pre-fixing the ordering of the training data based on perplexity scores from a pre-trained model performs the best and outperforms the default approach of randomly shuffling the training data every epoch.

Uniform random rotations are a useful primitive in applications such as fast Johnson-Lindenstrauss embeddings, kernel approximation, communication-efficient learning, and recent AI compression pipelines, but they are computationally expensive to generate and apply in high dimensions. A common practical replacement is repeated structured random rotations built from Walsh-Hadamard transforms and random sign diagonals. Applying the structured random rotation twice has been shown empirically to be useful, but the supporting theory is still limited. In this paper we study the approximation quality achieved when using this two-block structured Hadamard rotation. Our results are both positive and negative. On the positive side, we prove that every fixed coordinate of the two-block transform converges uniformly, over all inputs, to the corresponding coordinate of a uniformly rotated vector, with an explicit Kolmogorov-distance bound of order d^{-1/5}. On the negative side, we prove an explicit lower bound on the Wasserstein distance between the full vector distributions, showing that the two-block transform is not a globally accurate surrogate for a uniform random rotation in the worst case. For the extremal input used in the lower bound, we also prove a matching asymptotic upper bound, showing that the lower-bound scale is sharp for that input. Taken together, the results identify a clear separation between one-dimensional marginal behavior, where approximation improves with dimension, and full high-dimensional geometry, where a nonvanishing discrepancy remains. This provides a partial theoretical explanation for the empirical success of structured Hadamard rotations in some algorithms, while also clarifying the limitations of treating them as drop-in replacements for true uniform random rotations.

We investigate the expressive power of depth-2 bandlimited random neural networks. A random net is a neural network where the hidden layer parameters are frozen with random assignment, and only the output layer parameters are trained by loss minimization. Using random weights for a hidden layer is an effective method to avoid non-convex optimization in standard gradient descent learning. It has also been adopted in recent deep learning theories. Despite the well-known fact that a neural network is a universal approximator, in this study, we mathematically show that when hidden parameters are distributed in a bounded domain, the network may not achieve zero approximation error. In particular, we derive a new nontrivial approximation error lower bound. The proof utilizes the technique of ridgelet analysis, a harmonic analysis method designed for neural networks. This method is inspired by fundamental principles in classical signal processing, specifically the idea that signals with limited bandwidth may not always be able to perfectly recreate the original signal. We corroborate our theoretical results with various simulation studies, and generally, two main take-home messages are offered: (i) Not any distribution for selecting random weights is feasible to build a universal approximator; (ii) A suitable assignment of random weights exists but to some degree is associated with the complexity of the target function.

In shuffle privacy, each user sends a collection of randomized messages to a trusted shuffler, the shuffler randomly permutes these messages, and the resulting shuffled collection of messages must satisfy differential privacy. Prior work in this model has largely focused on protocols that use a single round of communication to compute algorithmic primitives like means, histograms, and counts. We present interactive shuffle protocols for stochastic convex optimization. Our protocols rely on a new noninteractive protocol for summing vectors of bounded ell_2 norm. By combining this sum subroutine with mini-batch stochastic gradient descent, accelerated gradient descent, and Nesterov's smoothing method, we obtain loss guarantees for a variety of convex loss functions that significantly improve on those of the local model and sometimes match those of the central model.

We define a probabilistic programming language for Gaussian random variables with a first-class exact conditioning construct. We give operational, denotational and equational semantics for this language, establishing convenient properties like exchangeability of conditions. Conditioning on equality of continuous random variables is nontrivial, as the exact observation may have probability zero; this is Borel's paradox. Using categorical formulations of conditional probability, we show that the good properties of our language are not particular to Gaussians, but can be derived from universal properties, thus generalizing to wider settings. We define the Cond construction, which internalizes conditioning as a morphism, providing general compositional semantics for probabilistic programming with exact conditioning.

Mehta and Panigrahi (2012) proposed Online Matching with Stochastic Rewards, which generalizes the Online Bipartite Matching problem of Karp, Vazirani, and Vazirani (1990) by associating the edges with success probabilities. This new feature captures the pay-per-click model in online advertising. Recently, Huang and Zhang (2020) studied this problem under the online primal dual framework using the Configuration Linear Program (LP), and got the best known competitive ratios of the Stochastic Balance algorithm. Their work suggests that the more expressive Configuration LP is more suitable for this problem than the Matching LP. This paper advances the theory of Configuration LP in two directions. Our technical contribution includes a characterization of the joint matching outcome of an offline vertex and all its neighbors. This characterization may be of independent interest, and is aligned with the spirit of Configuration LP. By contrast, previous analyses of Ranking generally focus on only one neighbor. Second, we designed a Stochastic Configuration LP that captures a stochastic benchmark proposed by Goyal and Udwani (2020), who used a Path-based LP. The Stochastic Configuration LP is smaller and simpler than the Path-based LP. Moreover, using the new LP we improved the competitive ratio of Stochastic Balance from 0.596 to 0.611 when the success probabilities are infinitesimal, and to 0.613 when the success probabilities are further equal.

This is the third paper of the CayleyPy project applying artificial intelligence to problems in group theory. We announce the first public release of CayleyPy, an open source Python library for computations with Cayley and Schreier graphs. Compared with systems such as GAP and Sage, CayleyPy handles much larger graphs and performs several orders of magnitude faster. Using CayleyPy we obtained about 200 new conjectures on Cayley and Schreier graphs, focused on diameters and growth. For many Cayley graphs of symmetric groups Sn we observe quasi polynomial diameter formulas: a small set of quadratic or linear polynomials indexed by n mod s. We conjecture that this is a general phenomenon, giving efficient diameter computation despite the problem being NP hard. We propose a refinement of the Babai type conjecture on diameters of Sn: n^2/2 + 4n upper bounds in the undirected case, compared to previous O(n^2) bounds. We also provide explicit generator families, related to involutions in a square with whiskers pattern, conjectured to maximize the diameter; search confirms this for all n up to 15. We further conjecture an answer to a question posed by V M Glushkov in 1968 on directed Cayley graphs generated by a cyclic shift and a transposition. For nilpotent groups we conjecture an improvement of J S Ellenberg's results on upper unitriangular matrices over Z/pZ, showing linear dependence of diameter on p. Moreover. Some conjectures are LLM friendly, naturally stated as sorting problems verifiable by algorithms or Python code. To benchmark path finding we created more than 10 Kaggle datasets. CayleyPy works with arbitrary permutation or matrix groups and includes over 100 predefined generators. Our growth computation code outperforms GAP and Sage up to 1000 times in speed and size.

We study the complexity of sampling from a distribution over all index subsets of the set {1,...,n} with the probability of a subset S proportional to the determinant of the submatrix L_S of some ntimes n p.s.d. matrix L, where L_S corresponds to the entries of L indexed by S. Known as a determinantal point process, this distribution is used in machine learning to induce diversity in subset selection. In practice, we often wish to sample multiple subsets S with small expected size k = E[|S|] ll n from a very large matrix L, so it is important to minimize the preprocessing cost of the procedure (performed once) as well as the sampling cost (performed repeatedly). For this purpose, we propose a new algorithm which, given access to L, samples exactly from a determinantal point process while satisfying the following two properties: (1) its preprocessing cost is n cdot poly(k), i.e., sublinear in the size of L, and (2) its sampling cost is poly(k), i.e., independent of the size of L. Prior to our results, state-of-the-art exact samplers required O(n^3) preprocessing time and sampling time linear in n or dependent on the spectral properties of L. We also give a reduction which allows using our algorithm for exact sampling from cardinality constrained determinantal point processes with ncdotpoly(k) time preprocessing.

The impact of randomness on model training is poorly understood. How do differences in data order and initialization actually manifest in the model, such that some training runs outperform others or converge faster? Furthermore, how can we interpret the resulting training dynamics and the phase transitions that characterize different trajectories? To understand the effect of randomness on the dynamics and outcomes of neural network training, we train models multiple times with different random seeds and compute a variety of metrics throughout training, such as the L_2 norm, mean, and variance of the neural network's weights. We then fit a hidden Markov model (HMM) over the resulting sequences of metrics. The HMM represents training as a stochastic process of transitions between latent states, providing an intuitive overview of significant changes during training. Using our method, we produce a low-dimensional, discrete representation of training dynamics on grokking tasks, image classification, and masked language modeling. We use the HMM representation to study phase transitions and identify latent "detour" states that slow down convergence.

Random Neural Networks (RNNs) are a class of Neural Networks (NNs) that can also be seen as a specific type of queuing network. They have been successfully used in several domains during the last 25 years, as queuing networks to analyze the performance of resource sharing in many engineering areas, as learning tools and in combinatorial optimization, where they are seen as neural systems, and also as models of neurological aspects of living beings. In this article we focus on their learning capabilities, and more specifically, we present a practical guide for using the RNN to solve supervised learning problems. We give a general description of these models using almost indistinctly the terminology of Queuing Theory and the neural one. We present the standard learning procedures used by RNNs, adapted from similar well-established improvements in the standard NN field. We describe in particular a set of learning algorithms covering techniques based on the use of first order and, then, of second order derivatives. We also discuss some issues related to these objects and present new perspectives about their use in supervised learning problems. The tutorial describes their most relevant applications, and also provides a large bibliography.

In reinforcement learning an agent interacts with the environment by taking actions and observing the next state and reward. When sampled probabilistically, these state transitions, rewards, and actions can all induce randomness in the observed long-term return. Traditionally, reinforcement learning algorithms average over this randomness to estimate the value function. In this paper, we build on recent work advocating a distributional approach to reinforcement learning in which the distribution over returns is modeled explicitly instead of only estimating the mean. That is, we examine methods of learning the value distribution instead of the value function. We give results that close a number of gaps between the theoretical and algorithmic results given by Bellemare, Dabney, and Munos (2017). First, we extend existing results to the approximate distribution setting. Second, we present a novel distributional reinforcement learning algorithm consistent with our theoretical formulation. Finally, we evaluate this new algorithm on the Atari 2600 games, observing that it significantly outperforms many of the recent improvements on DQN, including the related distributional algorithm C51.

Stochastic rounding (SR) is a probabilistic method used to round numbers to floating-point and fixed-point representations. In length n summation, the worst-case error of SR grows as n with high probability, unlike for standard modes, like round-to-nearest (RN), which grows as n. For this reason, the former is increasingly employed in large-scale, low-precision computations as an RN alternative. Additionally, SR alleviates stagnation, whereby relatively small summands are completely rounded off and do not contribute to the sum. We provide an update to [Croci et al., Roy. Soc. Open Sci. 9.3 (2022), pp. 1-25], a survey which discusses the development and use of SR between 1949 and 2022, citing over 100 references. Since then, there has been a surge of new research, and this update covers almost four years of further progress in applying, analysing, and implementing SR. Our main focus is limited-precision stochastic rounding, a new variant that fixes the precision of the random numbers used. We provide insights into industrial and numerical analysis activities surrounding SR, highlighting the next possible steps in making this rounding mode more widely available in hardware.

Many machine learning tasks such as multiple instance learning, 3D shape recognition, and few-shot image classification are defined on sets of instances. Since solutions to such problems do not depend on the order of elements of the set, models used to address them should be permutation invariant. We present an attention-based neural network module, the Set Transformer, specifically designed to model interactions among elements in the input set. The model consists of an encoder and a decoder, both of which rely on attention mechanisms. In an effort to reduce computational complexity, we introduce an attention scheme inspired by inducing point methods from sparse Gaussian process literature. It reduces the computation time of self-attention from quadratic to linear in the number of elements in the set. We show that our model is theoretically attractive and we evaluate it on a range of tasks, demonstrating the state-of-the-art performance compared to recent methods for set-structured data.

LLMs are widely used for code generation and mathematical reasoning tasks where they are required to generate structured output. They either need to reason about code, generate code for a given specification, or reason using programs of thought. The typical approach to code generation is to prompt the model and generate samples until an appropriate program is obtained. Within this process, sampling n programs from the language model requires n GPU compute-intensive generations which becomes prohibitively expensive for larger values of n. In this work, we address this limitation by exposing the LLM's distribution within the generated programs themselves. We propose a novel test-time framework we dub probabilistic programs of thought to obtain more samples from the model with fewer LLM generations. Given a program generated by a model and the associated next-token probabilities, we build a probabilistic program that compactly represents exponentially many deterministic programs. Since performing probabilistic reasoning in this probabilistic program is much cheaper, our approach allows sampling new programs without any additional GPU compute and little CPU overhead. We instantiate our approach on benchmarks for code generation, code understanding and mathematical reasoning and report improvements in performance with fewer generations from the LLM.

During the last decade, sampling-based path planning algorithms, such as Probabilistic RoadMaps (PRM) and Rapidly-exploring Random Trees (RRT), have been shown to work well in practice and possess theoretical guarantees such as probabilistic completeness. However, little effort has been devoted to the formal analysis of the quality of the solution returned by such algorithms, e.g., as a function of the number of samples. The purpose of this paper is to fill this gap, by rigorously analyzing the asymptotic behavior of the cost of the solution returned by stochastic sampling-based algorithms as the number of samples increases. A number of negative results are provided, characterizing existing algorithms, e.g., showing that, under mild technical conditions, the cost of the solution returned by broadly used sampling-based algorithms converges almost surely to a non-optimal value. The main contribution of the paper is the introduction of new algorithms, namely, PRM* and RRT*, which are provably asymptotically optimal, i.e., such that the cost of the returned solution converges almost surely to the optimum. Moreover, it is shown that the computational complexity of the new algorithms is within a constant factor of that of their probabilistically complete (but not asymptotically optimal) counterparts. The analysis in this paper hinges on novel connections between stochastic sampling-based path planning algorithms and the theory of random geometric graphs.

We study sampling algorithms for β-ensembles with time complexity less than cubic in the cardinality of the ensemble. Following Dumitriu & Edelman (2002), we see the ensemble as the eigenvalues of a random tridiagonal matrix, namely a random Jacobi matrix. First, we provide a unifying and elementary treatment of the tridiagonal models associated to the three classical Hermite, Laguerre and Jacobi ensembles. For this purpose, we use simple changes of variables between successive reparametrizations of the coefficients defining the tridiagonal matrix. Second, we derive an approximate sampler for the simulation of β-ensembles, and illustrate how fast it can be for polynomial potentials. This method combines a Gibbs sampler on Jacobi matrices and the diagonalization of these matrices. In practice, even for large ensembles, only a few Gibbs passes suffice for the marginal distribution of the eigenvalues to fit the expected theoretical distribution. When the conditionals in the Gibbs sampler can be simulated exactly, the same fast empirical convergence is observed for the fluctuations of the largest eigenvalue. Our experimental results support a conjecture by Krishnapur et al. (2016), that the Gibbs chain on Jacobi matrices of size N mixes in O(log(N)).

Can we learn more from data than existed in the generating process itself? Can new and useful information be constructed from merely applying deterministic transformations to existing data? Can the learnable content in data be evaluated without considering a downstream task? On these questions, Shannon information and Kolmogorov complexity come up nearly empty-handed, in part because they assume observers with unlimited computational capacity and fail to target the useful information content. In this work, we identify and exemplify three seeming paradoxes in information theory: (1) information cannot be increased by deterministic transformations; (2) information is independent of the order of data; (3) likelihood modeling is merely distribution matching. To shed light on the tension between these results and modern practice, and to quantify the value of data, we introduce epiplexity, a formalization of information capturing what computationally bounded observers can learn from data. Epiplexity captures the structural content in data while excluding time-bounded entropy, the random unpredictable content exemplified by pseudorandom number generators and chaotic dynamical systems. With these concepts, we demonstrate how information can be created with computation, how it depends on the ordering of the data, and how likelihood modeling can produce more complex programs than present in the data generating process itself. We also present practical procedures to estimate epiplexity which we show capture differences across data sources, track with downstream performance, and highlight dataset interventions that improve out-of-distribution generalization. In contrast to principles of model selection, epiplexity provides a theoretical foundation for data selection, guiding how to select, generate, or transform data for learning systems.

Experience replay is a key component in reinforcement learning for stabilizing learning and improving sample efficiency. Its typical implementation samples transitions with replacement from a replay buffer. In contrast, in supervised learning with a fixed dataset, it is a common practice to shuffle the dataset every epoch and consume data sequentially, which is called random reshuffling (RR). RR enjoys theoretically better convergence properties and has been shown to outperform with-replacement sampling empirically. To leverage the benefits of RR in reinforcement learning, we propose sampling methods that extend RR to experience replay, both in uniform and prioritized settings. We evaluate our sampling methods on Atari benchmarks, demonstrating their effectiveness in deep reinforcement learning.

The fastrerandomize R package provides hardware-accelerated tools for performing rerandomization and randomization testing in experimental research. Using a JAX backend, the package enables exact rerandomization inference even for large experiments with hundreds of billions of possible randomizations. Key functionalities include generating pools of acceptable rerandomizations based on covariate balance, conducting exact randomization tests, and performing pre-analysis evaluations to determine optimal rerandomization acceptance thresholds. Through batched processing and GPU acceleration, fastrerandomize achieves substantial performance gains compared to existing implementations, making previously intractable designs computationally feasible. The package therefore extends the randomization-based inference toolkit in R, allowing researchers to efficiently implement more stringent rerandomization designs and conduct valid inference even with large sample sizes or in high-dimensional settings.

Random masks define surprisingly effective sparse neural network models, as has been shown empirically. The resulting sparse networks can often compete with dense architectures and state-of-the-art lottery ticket pruning algorithms, even though they do not rely on computationally expensive prune-train iterations and can be drawn initially without significant computational overhead. We offer a theoretical explanation of how random masks can approximate arbitrary target networks if they are wider by a logarithmic factor in the inverse sparsity 1 / log(1/sparsity). This overparameterization factor is necessary at least for 3-layer random networks, which elucidates the observed degrading performance of random networks at higher sparsity. At moderate to high sparsity levels, however, our results imply that sparser networks are contained within random source networks so that any dense-to-sparse training scheme can be turned into a computationally more efficient sparse-to-sparse one by constraining the search to a fixed random mask. We demonstrate the feasibility of this approach in experiments for different pruning methods and propose particularly effective choices of initial layer-wise sparsity ratios of the random source network. As a special case, we show theoretically and experimentally that random source networks also contain strong lottery tickets.

Masked diffusion models (MDMs) generate text by iteratively selecting positions to unmask and then predicting tokens at those positions. Yet MDMs lack proper likelihood evaluation: the evidence lower bound (ELBO) is not only a loose bound on log-likelihood, but, as we show, is also computed under the training distribution rather than the test-time distribution. We resolve this within our DUEL framework, which unifies leading MDM sampling strategies that employ deterministic position selection. We prove that DUEL samplers admit exact likelihood computation under the test-time distribution -- giving MDMs proper likelihood, and hence proper perplexity, for the first time. This proper perplexity is the natural analogue of autoregressive perplexity and lets us revisit key questions about MDMs. MDMs are substantially better than previously thought: the MDM-autoregressive perplexity gap shrinks by up to 32% on in-domain data and 82% on zero-shot benchmarks. DUEL enables the first principled comparison of fast,parallel samplers across compute budgets -- an analysis impossible with the ELBO and unreliable with generative perplexity -- identifying a strong default method. Finally, oracle search over position orderings reveals MDMs can far surpass autoregressive models -- achieving 36.47 vs. 52.11 perplexity on AG News -- demonstrating the ceiling of MDM performance has not yet been reached.

We study the chromatic number of typical triangle-free graphs with Theta left( n^{3/2} (log n)^{1/2} right) edges and establish the width of the scaling window for the transitions from chi = 3 to chi = 4 and from chi = 4 to chi = 5. The transition from 3- to 4-colorability has scaling window of width Theta(n^{4/3} (log n)^{-1/3}). To prove this, we show a high probability equivalence of the 3-colorability of a random triangle-free graph at this density and the satisfiability of an instance of bipartite random 2-SAT, for which we establish the width of the scaling window following the techniques of Bollob{\'a}s, Borgs, Chayes, Kim, and Wilson. The transition from 4- to 5-colorability has scaling window of width Theta(n^{3/2} (log n)^{-1/2}). To prove this, we show a high probability equivalence of the 4-colorability of a random triangle-free graph at this density and the simultaneous 2-colorability of two independent Erdos--R\'enyi random graphs. For this transition, we also establish the limiting probability of 4-colorability inside the scaling window.

While plan-and-infill decoding in Masked Diffusion Models (MDMs) shows promise for mathematical and code reasoning, performance remains highly sensitive to slot infilling order, often yielding substantial output variance. We introduce McDiffuSE, a framework that formulates slot selection as decision making and optimises infilling orders through Monte Carlo Tree Search (MCTS). McDiffuSE uses look-ahead simulations to evaluate partial completions before commitment, systematically exploring the combinatorial space of generation orders. Experiments show an average improvement of 3.2% over autoregressive baselines and 8.0% over baseline plan-and-infill, with notable gains of 19.5% on MBPP and 4.9% on MATH500. Our analysis reveals that while McDiffuSE predominantly follows sequential ordering, incorporating non-sequential generation is essential for maximising performance. We observe that larger exploration constants, rather than increased simulations, are necessary to overcome model confidence biases and discover effective orderings. These findings establish MCTS-based planning as an effective approach for enhancing generation quality in MDMs.

Sequence models such as transformers require inputs to be represented as one-dimensional sequences. In vision, this typically involves flattening images using a fixed row-major (raster-scan) order. While full self-attention is permutation-equivariant, modern long-sequence transformers increasingly rely on architectural approximations that break this invariance and introduce sensitivity to patch ordering. We show that patch order significantly affects model performance in such settings, with simple alternatives like column-major or Hilbert curves yielding notable accuracy shifts. Motivated by this, we propose REOrder, a two-stage framework for discovering task-optimal patch orderings. First, we derive an information-theoretic prior by evaluating the compressibility of various patch sequences. Then, we learn a policy over permutations by optimizing a Plackett-Luce policy using REINFORCE. This approach enables efficient learning in a combinatorial permutation space. REOrder improves top-1 accuracy over row-major ordering on ImageNet-1K by up to 3.01% and Functional Map of the World by 13.35%.

This paper is an extended and reworked version of a short course given by the author at ''Uzbekistan-Ukrainian readings in stochastic processes'', Tashkent-Kyiv, 2022, and was prepared for a special issue of ''Theory of stochastic processes'', devoted to publishing lecture notes from the aforementioned workshop. The survey is devoted to operator splitting methods in the abstract formulation and their applications in probability. While the survey is focused on multiplicative methods, the BCH formula is used to discuss exponential splitting methods and a short informal introduction to additive splitting is presented. We introduce frameworks and available deterministic and probabilistic results and concentrate on constructing a wide picture of the field of operator splitting methods, providing a rigorous description in the setting of abstract Cauchy problems and an informal discussion for further and parallel advances. Some limitations and common difficulties are listed, as well as examples of works that provide solutions or hints. No new results are given. The bibliography contains illustrative deterministic examples and a selection of probability-related works.

Completely randomized experiments, originally developed by Fisher and Neyman in the 1930s, are still widely used in practice, even in online experimentation. However, such designs are of limited value for answering standard questions in marketplaces, where multiple populations of agents interact strategically, leading to complex patterns of spillover effects. In this paper, we derive the finite-sample properties of tractable estimators for "Simple Multiple Randomization Designs" (SMRDs), a new class of experimental designs which account for complex spillover effects in randomized experiments. Our derivations are obtained under a natural and general form of cross-unit interference, which we call "local interference". We discuss the estimation of main effects, direct effects, and spillovers, and present associated central limit theorems.

Over the past few years, the acceleration of computing resources and research in deep learning has led to significant practical successes in a range of tasks, including in particular in computer vision. Building on these advances, reinforcement learning has also seen a leap forward with the emergence of agents capable of making decisions directly from visual observations. Despite these successes, the over-parametrization of neural architectures leads to memorization of the data used during training and thus to a lack of generalization. Reinforcement learning agents based on visual inputs also suffer from this phenomenon by erroneously correlating rewards with unrelated visual features such as background elements. To alleviate this problem, we introduce a new regularization technique consisting of channel-consistent local permutations (CLOP) of the feature maps. The proposed permutations induce robustness to spatial correlations and help prevent overfitting behaviors in RL. We demonstrate, on the OpenAI Procgen Benchmark, that RL agents trained with the CLOP method exhibit robustness to visual changes and better generalization properties than agents trained using other state-of-the-art regularization techniques. We also demonstrate the effectiveness of CLOP as a general regularization technique in supervised learning.

The Markov numbers are positive integers appearing as solutions to the Diophantine equation x^2 + y^2 + z^2 = 3xyz. These numbers are very well-studied and have many combinatorial properties, as well as being the source of the long-standing unicity conjecture. In 2018, Canakc{\i} and Schiffler showed that the Markov number m_{a{b}} is the number of perfect matchings of a certain snake graph corresponding to the Christoffel path from (0,0) to (a,b). Based on this correspondence, Schiffler in 2023 introduced two orderings on lattice paths. For any path omega, associate a snake graph G(omega) and a continued fraction g(omega). The ordering <_M is given by the number of perfect matchings on G(omega), and the ordering <_L is given by the Lagrange number of g(omega). In this work, we settle two conjectures of Schiffler. First, we show that the path omega(a,b) = RRcdots R UU cdots U is the unique maximum over all lattice paths from (0,0) to (a,b) with respect to both orderings <_M and <_L. We then use this result to prove that sup L(omega) over all lattice paths is exactly 1+sqrt5.

The Brownian sphere is a random metric space, homeomorphic to the two-dimensional sphere, which arises as the universal scaling limit of many types of random planar maps. The direct construction of the Brownian sphere is via a continuous analogue of the Cori--Vauquelin--Schaeffer (CVS) bijection. The CVS bijection maps labeled trees to planar maps, and the continuous version maps Aldous' continuum random tree with Brownian labels (the Brownian snake) to the Brownian sphere. In this work, we describe the inverse of the continuous CVS bijection, by constructing the Brownian snake as a measurable function of the Brownian sphere. Special care is needed to work with the orientation of the Brownian sphere.

This paper provides a non-asymptotic analysis of linear stochastic approximation (LSA) algorithms with fixed stepsize. This family of methods arises in many machine learning tasks and is used to obtain approximate solutions of a linear system Atheta = b for which A and b can only be accessed through random estimates {({bf A}_n, {bf b}_n): n in N^*}. Our analysis is based on new results regarding moments and high probability bounds for products of matrices which are shown to be tight. We derive high probability bounds on the performance of LSA under weaker conditions on the sequence {({bf A}_n, {bf b}_n): n in N^*} than previous works. However, in contrast, we establish polynomial concentration bounds with order depending on the stepsize. We show that our conclusions cannot be improved without additional assumptions on the sequence of random matrices {{bf A}_n: n in N^*}, and in particular that no Gaussian or exponential high probability bounds can hold. Finally, we pay a particular attention to establishing bounds with sharp order with respect to the number of iterations and the stepsize and whose leading terms contain the covariance matrices appearing in the central limit theorems.

Graph generation has been dominated by autoregressive models due to their simplicity and effectiveness, despite their sensitivity to ordering. Yet diffusion models have garnered increasing attention, as they offer comparable performance while being permutation-invariant. Current graph diffusion models generate graphs in a one-shot fashion, but they require extra features and thousands of denoising steps to achieve optimal performance. We introduce PARD, a Permutation-invariant Auto Regressive Diffusion model that integrates diffusion models with autoregressive methods. PARD harnesses the effectiveness and efficiency of the autoregressive model while maintaining permutation invariance without ordering sensitivity. Specifically, we show that contrary to sets, elements in a graph are not entirely unordered and there is a unique partial order for nodes and edges. With this partial order, PARD generates a graph in a block-by-block, autoregressive fashion, where each block's probability is conditionally modeled by a shared diffusion model with an equivariant network. To ensure efficiency while being expressive, we further propose a higher-order graph transformer, which integrates transformer with PPGN. Like GPT, we extend the higher-order graph transformer to support parallel training of all blocks. Without any extra features, PARD achieves state-of-the-art performance on molecular and non-molecular datasets, and scales to large datasets like MOSES containing 1.9M molecules.

In this paper, we consider the problem of sleeping bandits with stochastic action sets and adversarial rewards. In this setting, in contrast to most work in bandits, the actions may not be available at all times. For instance, some products might be out of stock in item recommendation. The best existing efficient (i.e., polynomial-time) algorithms for this problem only guarantee an O(T^{2/3}) upper-bound on the regret. Yet, inefficient algorithms based on EXP4 can achieve O(T). In this paper, we provide a new computationally efficient algorithm inspired by EXP3 satisfying a regret of order O(T) when the availabilities of each action i in cA are independent. We then study the most general version of the problem where at each round available sets are generated from some unknown arbitrary distribution (i.e., without the independence assumption) and propose an efficient algorithm with O(2^K T) regret guarantee. Our theoretical results are corroborated with experimental evaluations.

We show that classifiers trained with random region proposals achieve state-of-the-art Open-world Object Detection (OWOD): they can not only maintain the accuracy of the known objects (w/ training labels), but also considerably improve the recall of unknown ones (w/o training labels). Specifically, we propose RandBox, a Fast R-CNN based architecture trained on random proposals at each training iteration, surpassing existing Faster R-CNN and Transformer based OWOD. Its effectiveness stems from the following two benefits introduced by randomness. First, as the randomization is independent of the distribution of the limited known objects, the random proposals become the instrumental variable that prevents the training from being confounded by the known objects. Second, the unbiased training encourages more proposal explorations by using our proposed matching score that does not penalize the random proposals whose prediction scores do not match the known objects. On two benchmarks: Pascal-VOC/MS-COCO and LVIS, RandBox significantly outperforms the previous state-of-the-art in all metrics. We also detail the ablations on randomization and loss designs. Codes are available at https://github.com/scuwyh2000/RandBox.

Learning neural subset selection tasks, such as compound selection in AI-aided drug discovery, have become increasingly pivotal across diverse applications. The existing methodologies in the field primarily concentrate on constructing models that capture the relationship between utility function values and subsets within their respective supersets. However, these approaches tend to overlook the valuable information contained within the superset when utilizing neural networks to model set functions. In this work, we address this oversight by adopting a probabilistic perspective. Our theoretical findings demonstrate that when the target value is conditioned on both the input set and subset, it is essential to incorporate an invariant sufficient statistic of the superset into the subset of interest for effective learning. This ensures that the output value remains invariant to permutations of the subset and its corresponding superset, enabling identification of the specific superset from which the subset originated. Motivated by these insights, we propose a simple yet effective information aggregation module designed to merge the representations of subsets and supersets from a permutation invariance perspective. Comprehensive empirical evaluations across diverse tasks and datasets validate the enhanced efficacy of our approach over conventional methods, underscoring the practicality and potency of our proposed strategies in real-world contexts.

Given a set of K probability densities, we consider the multimarginal generative modeling problem of learning a joint distribution that recovers these densities as marginals. The structure of this joint distribution should identify multi-way correspondences among the prescribed marginals. We formalize an approach to this task within a generalization of the stochastic interpolant framework, leading to efficient learning algorithms built upon dynamical transport of measure. Our generative models are defined by velocity and score fields that can be characterized as the minimizers of simple quadratic objectives, and they are defined on a simplex that generalizes the time variable in the usual dynamical transport framework. The resulting transport on the simplex is influenced by all marginals, and we show that multi-way correspondences can be extracted. The identification of such correspondences has applications to style transfer, algorithmic fairness, and data decorruption. In addition, the multimarginal perspective enables an efficient algorithm for reducing the dynamical transport cost in the ordinary two-marginal setting. We demonstrate these capacities with several numerical examples.

In this work we develop a new method, named Sub-graph Permutation Equivariant Networks (SPEN), which provides a framework for building graph neural networks that operate on sub-graphs, while using a base update function that is permutation equivariant, that are equivariant to a novel choice of automorphism group. Message passing neural networks have been shown to be limited in their expressive power and recent approaches to over come this either lack scalability or require structural information to be encoded into the feature space. The general framework presented here overcomes the scalability issues associated with global permutation equivariance by operating more locally on sub-graphs. In addition, through operating on sub-graphs the expressive power of higher-dimensional global permutation equivariant networks is improved; this is due to fact that two non-distinguishable graphs often contain distinguishable sub-graphs. Furthermore, the proposed framework only requires a choice of k-hops for creating ego-network sub-graphs and a choice of representation space to be used for each layer, which makes the method easily applicable across a range of graph based domains. We experimentally validate the method on a range of graph benchmark classification tasks, demonstrating statistically indistinguishable results from the state-of-the-art on six out of seven benchmarks. Further, we demonstrate that the use of local update functions offers a significant improvement in GPU memory over global methods.

Verifying the provenance of content is crucial to the function of many organizations, e.g., educational institutions, social media platforms, firms, etc. This problem is becoming increasingly difficult as text generated by Large Language Models (LLMs) becomes almost indistinguishable from human-generated content. In addition, many institutions utilize in-house LLMs and want to ensure that external, non-sanctioned LLMs do not produce content within the institution. In this paper, we answer the following question: Given a piece of text, can we identify whether it was produced by LLM A or B (where B can be a human)? We model LLM-generated text as a sequential stochastic process with complete dependence on history and design zero-shot statistical tests to distinguish between (i) the text generated by two different sets of LLMs A (in-house) and B (non-sanctioned) and also (ii) LLM-generated and human-generated texts. We prove that the type I and type II errors for our tests decrease exponentially in the text length. In designing our tests, we derive concentration inequalities on the difference between log-perplexity and the average entropy of the string under A. Specifically, for a given string, we demonstrate that if the string is generated by A, the log-perplexity of the string under A converges to the average entropy of the string under A, except with an exponentially small probability in string length. We also show that if B generates the text, except with an exponentially small probability in string length, the log-perplexity of the string under A converges to the average cross-entropy of B and A. Lastly, we present preliminary experimental results to support our theoretical results. By enabling guaranteed (with high probability) finding of the origin of harmful LLM-generated text with arbitrary size, we can help combat misinformation.

Existing combinatorial search methods are often complex and require some level of expertise. This work introduces a simple and efficient deep learning method for solving combinatorial problems with a predefined goal, represented by Rubik's Cube. We demonstrate that, for such problems, training a deep neural network on random scrambles branching from the goal state is sufficient to achieve near-optimal solutions. When tested on Rubik's Cube, 15 Puzzle, and 7times7 Lights Out, our method outperformed the previous state-of-the-art method DeepCubeA, improving the trade-off between solution optimality and computational cost, despite significantly less training data. Furthermore, we investigate the scaling law of our Rubik's Cube solver with respect to model size and training data volume.

Neural networks for image recognition have evolved through extensive manual design from simple chain-like models to structures with multiple wiring paths. The success of ResNets and DenseNets is due in large part to their innovative wiring plans. Now, neural architecture search (NAS) studies are exploring the joint optimization of wiring and operation types, however, the space of possible wirings is constrained and still driven by manual design despite being searched. In this paper, we explore a more diverse set of connectivity patterns through the lens of randomly wired neural networks. To do this, we first define the concept of a stochastic network generator that encapsulates the entire network generation process. Encapsulation provides a unified view of NAS and randomly wired networks. Then, we use three classical random graph models to generate randomly wired graphs for networks. The results are surprising: several variants of these random generators yield network instances that have competitive accuracy on the ImageNet benchmark. These results suggest that new efforts focusing on designing better network generators may lead to new breakthroughs by exploring less constrained search spaces with more room for novel design.

We introduce OpenRAND, a C++17 library aimed at facilitating reproducible scientific research through the generation of statistically robust and yet replicable random numbers. OpenRAND accommodates single and multi-threaded applications on CPUs and GPUs and offers a simplified, user-friendly API that complies with the C++ standard's random number engine interface. It is portable: it functions seamlessly as a lightweight, header-only library, making it adaptable to a wide spectrum of software and hardware platforms. It is statistically robust: a suite of built-in tests ensures no pattern exists within single or multiple streams. Despite the simplicity and portability, it is remarkably performant-matching and sometimes even outperforming native libraries by a significant margin. Our tests, including a Brownian walk simulation, affirm its reproducibility and highlight its computational efficiency, outperforming CUDA's cuRAND by up to 1.8 times.

We present ModularSubsetSelection (MSS), a new algorithm for locally differentially private (LDP) frequency estimation. Given a universe of size k and n users, our varepsilon-LDP mechanism encodes each input via a Residue Number System (RNS) over ell pairwise-coprime moduli m_0, ldots, m_{ell-1}, and reports a randomly chosen index j in [ell] along with the perturbed residue using the statistically optimal SubsetSelection (SS) (Wang et al. 2016). This design reduces the user communication cost from Θbigl(ωlog_2(k/ω)bigr) bits required by standard SS (with ωapprox k/(e^varepsilon+1)) down to lceil log_2 ell rceil + lceil log_2 m_j rceil bits, where m_j < k. Server-side decoding runs in Θ(n + r k ell) time, where r is the number of LSMR (Fong and Saunders 2011) iterations. In practice, with well-conditioned moduli (i.e., constant r and ell = Θ(log k)), this becomes Θ(n + k log k). We prove that MSS achieves worst-case MSE within a constant factor of state-of-the-art protocols such as SS and ProjectiveGeometryResponse (PGR) (Feldman et al. 2022) while avoiding the algebraic prerequisites and dynamic-programming decoder required by PGR. Empirically, MSS matches the estimation accuracy of SS, PGR, and RAPPOR (Erlingsson, Pihur, and Korolova 2014) across realistic (k, varepsilon) settings, while offering faster decoding than PGR and shorter user messages than SS. Lastly, by sampling from multiple moduli and reporting only a single perturbed residue, MSS achieves the lowest reconstruction-attack success rate among all evaluated LDP protocols.

Large language models (LLMs) are now widely used to evaluate the quality of text, a field commonly referred to as LLM-as-a-judge. While prior works mainly focus on point-wise and pair-wise evaluation paradigms. Rubric-based evaluation, where LLMs select a score from multiple rubrics, has received less analysis. In this work, we show that rubric-based evaluation implicitly resembles a multi-choice setting and therefore has position bias: LLMs prefer score options appearing at specific positions in the rubric list. Through controlled experiments across multiple models and datasets, we demonstrate consistent position bias. To mitigate this bias, we propose a balanced permutation strategy that evenly distributes each score option across positions. We show that aggregating scores across balanced permutations not only reveals latent position bias, but also improves correlation between the LLM-as-a-Judge and human. Our results suggest that rubric-based LLM-as-a-Judge is not inherently point-wise and that simple permutation-based calibration can substantially improve its reliability.

Combinatorial optimization (CO) is naturally discrete, making machine learning based on differentiable optimization inapplicable. Karalias & Loukas (2020) adapted the probabilistic method to incorporate CO into differentiable optimization. Their work ignited the research on unsupervised learning for CO, composed of two main components: probabilistic objectives and derandomization. However, each component confronts unique challenges. First, deriving objectives under various conditions (e.g., cardinality constraints and minimum) is nontrivial. Second, the derandomization process is underexplored, and the existing derandomization methods are either random sampling or naive rounding. In this work, we aim to tackle prevalent (i.e., commonly involved) conditions in unsupervised CO. First, we concretize the targets for objective construction and derandomization with theoretical justification. Then, for various conditions commonly involved in different CO problems, we derive nontrivial objectives and derandomization to meet the targets. Finally, we apply the derivations to various CO problems. Via extensive experiments on synthetic and real-world graphs, we validate the correctness of our derivations and show our empirical superiority w.r.t. both optimization quality and speed.

We present an end-to-end and practical randomness amplification and privatisation protocol based on Bell tests. This allows the building of device-independent random number generators which output (near-)perfectly unbiased and private numbers, even if using an uncharacterised quantum device potentially built by an adversary. Our generation rates are linear in the repetition rate of the quantum device and the classical randomness post-processing has quasi-linear complexity - making it efficient on a standard personal laptop. The statistical analysis is also tailored for real-world quantum devices. Our protocol is then showcased on several different quantum computers. Although not purposely built for the task, we show that quantum computers can run faithful Bell tests by adding minimal assumptions. In this semi-device-independent manner, our protocol generates (near-)perfectly unbiased and private random numbers on today's quantum computers.

The lottery ticket hypothesis (Frankle and Carbin, 2018), states that a randomly-initialized network contains a small subnetwork such that, when trained in isolation, can compete with the performance of the original network. We prove an even stronger hypothesis (as was also conjectured in Ramanujan et al., 2019), showing that for every bounded distribution and every target network with bounded weights, a sufficiently over-parameterized neural network with random weights contains a subnetwork with roughly the same accuracy as the target network, without any further training.

Multi-layer perceptron (MLP) is a fundamental component of deep learning that has been extensively employed for various problems. However, recent empirical successes in MLP-based architectures, particularly the progress of the MLP-Mixer, have revealed that there is still hidden potential in improving MLPs to achieve better performance. In this study, we reveal that the MLP-Mixer works effectively as a wide MLP with certain sparse weights. Initially, we clarify that the mixing layer of the Mixer has an effective expression as a wider MLP whose weights are sparse and represented by the Kronecker product. This expression naturally defines a permuted-Kronecker (PK) family, which can be regarded as a general class of mixing layers and is also regarded as an approximation of Monarch matrices. Subsequently, because the PK family effectively constitutes a wide MLP with sparse weights, one can apply the hypothesis proposed by Golubeva, Neyshabur and Gur-Ari (2021) that the prediction performance improves as the width (sparsity) increases when the number of weights is fixed. We empirically verify this hypothesis by maximizing the effective width of the MLP-Mixer, which enables us to determine the appropriate size of the mixing layers quantitatively.

In this paper, we revisit one of the simplest problems in data structures: the task of inserting elements into an open-addressed hash table so that elements can later be retrieved with as few probes as possible. We show that, even without reordering elements over time, it is possible to construct a hash table that achieves far better expected search complexities (both amortized and worst-case) than were previously thought possible. Along the way, we disprove the central conjecture left by Yao in his seminal paper ``Uniform Hashing is Optimal''. All of our results come with matching lower bounds.

The binary sum-of-digits function s returns the number of ones in the binary expansion of a nonnegative integer. Cusick's Hamming weight conjecture states that, for all integers tgeq 0, the set of nonnegative integers n such that s(n+t)geq s(n) has asymptotic density strictly larger than 1/2. We are concerned with the block-additive function r returning the number of (overlapping) occurrences of the block 11 in the binary expansion of n. The main result of this paper is a central limit-type theorem for the difference r(n+t)-r(n): the corresponding probability function is uniformly close to a Gaussian, where the uniform error tends to 0 as the number of blocks of ones in the binary expansion of t tends to infty.

This article introduces Regression Discontinuity Design (RDD) with Distribution-Valued Outcomes (R3D), extending the standard RDD framework to settings where the outcome is a distribution rather than a scalar. Such settings arise when treatment is assigned at a higher level of aggregation than the outcome-for example, when a subsidy is allocated based on a firm-level revenue cutoff while the outcome of interest is the distribution of employee wages within the firm. Since standard RDD methods cannot accommodate such two-level randomness, I propose a novel approach based on random distributions. The target estimand is a "local average quantile treatment effect", which averages across random quantiles. To estimate this target, I introduce two related approaches: one that extends local polynomial regression to random quantiles and another based on local Fr\'echet regression, a form of functional regression. For both estimators, I establish asymptotic normality and develop uniform, debiased confidence bands together with a data-driven bandwidth selection procedure. Simulations validate these theoretical properties and show existing methods to be biased and inconsistent in this setting. I then apply the proposed methods to study the effects of gubernatorial party control on within-state income distributions in the US, using a close-election design. The results suggest a classic equality-efficiency tradeoff under Democratic governorship, driven by reductions in income at the top of the distribution.

We present a new method for constructing Hadamard matrices that combines transformer neural networks with local search in the PatternBoost framework. Our approach is designed for extremely sparse combinatorial search problems and is particularly effective for Hadamard matrices of Goethals--Seidel type, where Fourier methods permit fast scoring and optimisation. For orders between 100 and 200, it produces large numbers of inequivalent Hadamard matrices, and for larger orders, it succeeds where local search from random initialisation fails. The largest example found by our method has order 252. In addition to these new constructions, our experiments reveal that the transformer can discover and exploit useful hidden symmetry in the search space.

Consider a random uniform sample of n points in a compact region A of Euclidean d-space, d geq 2, with a smooth or (when d=2) polygonal boundary. Fix k bf N. Let T_{n,k} be the threshold r at which the geometric graph on these n vertices with distance parameter r becomes k-connected. We show that if d=2 then n (pi/|A|) T_{n,1}^2 - log n is asymptotically standard Gumbel. For (d,k) neq (2,1), it is n (theta_d/|A|) T_{n,k}^d - (2-2/d) log n - (4-2k-2/d) log log n that converges in distribution to a nondegenerate limit, where theta_d is the volume of the unit ball. The limit is Gumbel with scale parameter 2 except when (d,k)=(2,2) where the limit is two component extreme value distributed. The different cases reflect the fact that boundary effects are more more important in some cases than others. We also give similar results for the largest k-nearest neighbour link U_{n,k} in the sample, and show T_{n,k}=U_{n,k} with high probability. We provide estimates on rates of convergence and give similar results for Poisson samples in A. Finally, we give similar results even for non-uniform samples, with a less explicit sequence of centring constants.

Diffusion language models have emerged as a promising approach for text generation. One would naturally expect this method to be an efficient replacement for autoregressive models since multiple tokens can be sampled in parallel during each diffusion step. However, its efficiency-accuracy trade-off is not yet well understood. In this paper, we present a rigorous theoretical analysis of a widely used type of diffusion language model, the Masked Diffusion Model (MDM), and find that its effectiveness heavily depends on the target evaluation metric. Under mild conditions, we prove that when using perplexity as the metric, MDMs can achieve near-optimal perplexity in sampling steps regardless of sequence length, demonstrating that efficiency can be achieved without sacrificing performance. However, when using the sequence error rate--which is important for understanding the "correctness" of a sequence, such as a reasoning chain--we show that the required sampling steps must scale linearly with sequence length to obtain "correct" sequences, thereby eliminating MDM's efficiency advantage over autoregressive models. Our analysis establishes the first theoretical foundation for understanding the benefits and limitations of MDMs. All theoretical findings are supported by empirical studies.

Inspired by the fact that the neural network, as the mainstream for machine learning, has brought successes in many application areas, here we propose to use this approach for decoding hidden correlation among pseudo-random data and predicting events accordingly. With a simple neural network structure and a typical training procedure, we demonstrate the learning and prediction power of the neural network in extremely random environment. Finally, we postulate that the high sensitivity and efficiency of the neural network may allow to critically test if there could be any fundamental difference between quantum randomness and pseudo randomness, which is equivalent to the question: Does God play dice?

Designing biological sequences is an important challenge that requires satisfying complex constraints and thus is a natural problem to address with deep generative modeling. Diffusion generative models have achieved considerable success in many applications. Score-based generative stochastic differential equations (SDE) model is a continuous-time diffusion model framework that enjoys many benefits, but the originally proposed SDEs are not naturally designed for modeling discrete data. To develop generative SDE models for discrete data such as biological sequences, here we introduce a diffusion process defined in the probability simplex space with stationary distribution being the Dirichlet distribution. This makes diffusion in continuous space natural for modeling discrete data. We refer to this approach as Dirchlet diffusion score model. We demonstrate that this technique can generate samples that satisfy hard constraints using a Sudoku generation task. This generative model can also solve Sudoku, including hard puzzles, without additional training. Finally, we applied this approach to develop the first human promoter DNA sequence design model and showed that designed sequences share similar properties with natural promoter sequences.

Entanglement percolation aims at generating maximal entanglement between any two nodes of a quantum network by utilizing strategies based solely on local operations and classical communication between the nodes. As it happens in classical percolation theory, the topology of the network is crucial, but also the entanglement shared between the nodes of the network. In a network of identically partially entangled states, the network topology determines the minimum entanglement needed for percolation. In this work, we generalize the protocol to scenarios where the initial entanglement shared between each two nodes of the network is not the same but has some randomness. In such cases, we find that for classical entanglement percolation, only the average initial entanglement is relevant. In contrast, the quantum entanglement percolation protocol generally performs worse under these more realistic conditions.

The question of whether there exists a finite group of order at least three in which every element except one is a commutator has remained unresolved in group theory. In this article, we address this open problem by developing an algorithmic approach that leverages several group theoretic properties of such groups. Specifically, we utilize a result of Frobenius and various necessary properties of such groups, combined with Plesken and Holt's extensive enumeration of finite perfect groups, to systematically examine all finite groups up to a certain order for the desired property. The computational core of our work is implemented using the computer system GAP (Groups, Algorithms, and Programming). We discover two nonisomorphic groups of order 368,640 that exhibit the desired property. Our investigation also establishes that this order is the minimum order for such a group to exist. As a result, this study provides a positive answer to Problem 17.76 in the Kourovka Notebook. In addition to the algorithmic framework, this paper provides a structural description of one of the two groups found.

Higher-order probabilistic programming languages allow programmers to write sophisticated models in machine learning and statistics in a succinct and structured way, but step outside the standard measure-theoretic formalization of probability theory. Programs may use both higher-order functions and continuous distributions, or even define a probability distribution on functions. But standard probability theory does not handle higher-order functions well: the category of measurable spaces is not cartesian closed. Here we introduce quasi-Borel spaces. We show that these spaces: form a new formalization of probability theory replacing measurable spaces; form a cartesian closed category and so support higher-order functions; form a well-pointed category and so support good proof principles for equational reasoning; and support continuous probability distributions. We demonstrate the use of quasi-Borel spaces for higher-order functions and probability by: showing that a well-known construction of probability theory involving random functions gains a cleaner expression; and generalizing de Finetti's theorem, that is a crucial theorem in probability theory, to quasi-Borel spaces.

Quantum circuits -- built from local unitary gates and local measurements -- are a new playground for quantum many-body physics and a tractable setting to explore universal collective phenomena far-from-equilibrium. These models have shed light on longstanding questions about thermalization and chaos, and on the underlying universal dynamics of quantum information and entanglement. In addition, such models generate new sets of questions and give rise to phenomena with no traditional analog, such as new dynamical phases in quantum systems that are monitored by an external observer. Quantum circuit dynamics is also topical in view of experimental progress in building digital quantum simulators that allow control of precisely these ingredients. Randomness in the circuit elements allows a high level of theoretical control, with a key theme being mappings between real-time quantum dynamics and effective classical lattice models or dynamical processes. Many of the universal phenomena that can be identified in this tractable setting apply to much wider classes of more structured many-body dynamics.

Randomly initialized dense networks contain subnetworks that achieve high accuracy without weight learning--strong lottery tickets (SLTs). Recently, Gadhikar et al. (2023) demonstrated that SLTs could also be found within a randomly pruned source network. This phenomenon can be exploited to further compress the small memory size required by SLTs. However, their method is limited to SLTs that are even sparser than the source, leading to worse accuracy due to unintentionally high sparsity. This paper proposes a method for reducing the SLT memory size without restricting the sparsity of the SLTs that can be found. A random subset of the initial weights is frozen by either permanently pruning them or locking them as a fixed part of the SLT, resulting in a smaller model size. Experimental results show that Edge-Popup (Ramanujan et al., 2020; Sreenivasan et al., 2022) finds SLTs with better accuracy-to-model size trade-off within frozen networks than within dense or randomly pruned source networks. In particular, freezing 70% of a ResNet on ImageNet provides 3.3 times compression compared to the SLT found within a dense counterpart, raises accuracy by up to 14.12 points compared to the SLT found within a randomly pruned counterpart, and offers a better accuracy-model size trade-off than both.

Suppose Alice trains an open-weight language model and Bob uses a blackbox derivative of Alice's model to produce text. Can Alice prove that Bob is using her model, either by querying Bob's derivative model (query setting) or from the text alone (observational setting)? We formulate this question as an independence testing problem--in which the null hypothesis is that Bob's model or text is independent of Alice's randomized training run--and investigate it through the lens of palimpsestic memorization in language models: models are more likely to memorize data seen later in training, so we can test whether Bob is using Alice's model using test statistics that capture correlation between Bob's model or text and the ordering of training examples in Alice's training run. If Alice has randomly shuffled her training data, then any significant correlation amounts to exactly quantifiable statistical evidence against the null hypothesis, regardless of the composition of Alice's training data. In the query setting, we directly estimate (via prompting) the likelihood Bob's model gives to Alice's training examples and order; we correlate the likelihoods of over 40 fine-tunes of various Pythia and OLMo base models ranging from 1B to 12B parameters with the base model's training data order, achieving a p-value on the order of at most 1e-8 in all but six cases. In the observational setting, we try two approaches based on estimating 1) the likelihood of Bob's text overlapping with spans of Alice's training examples and 2) the likelihood of Bob's text with respect to different versions of Alice's model we obtain by repeating the last phase (e.g., 1%) of her training run on reshuffled data. The second approach can reliably distinguish Bob's text from as little as a few hundred tokens; the first does not involve any retraining but requires many more tokens (several hundred thousand) to achieve high power.