DPrivBench: Benchmarking LLMs' Reasoning for Differential Privacy
Paper • 2604.15851 • Published
question string | label int64 | function_id int64 | function string | function_sens string | question_id int64 |
|---|---|---|---|---|---|
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n x_i$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\varepsilon)))$. $Gumbel(t)$ stands for the... | 1 | 1 | f(X) = \sum_{i=1}^n x_i | 1 | 1 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n x_i$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\cdot(\varepsilon))))$. $Gumbel(t)$ stands... | 0 | 1 | f(X) = \sum_{i=1}^n x_i | 1 | 2 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \prod_{i=1}^n x_i$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\varepsilon)))$. $Gumbel(t)$ stands for th... | 1 | 2 | f(X) = \prod_{i=1}^n x_i | 1 | 3 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \prod_{i=1}^n x_i$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\cdot(\varepsilon))))$. $Gumbel(t)$ stand... | 0 | 2 | f(X) = \prod_{i=1}^n x_i | 1 | 4 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \max_{1 \le i \le n} x_i$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\varepsilon)))$. $Gumbel(t)$ stands... | 1 | 3 | f(X) = \max_{1 \le i \le n} x_i | 1 | 5 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \max_{1 \le i \le n} x_i$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\cdot(\varepsilon))))$. $Gumbel(t)... | 0 | 3 | f(X) = \max_{1 \le i \le n} x_i | 1 | 6 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \min_{1 \le i \le n} x_i$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\varepsilon)))$. $Gumbel(t)$ stands... | 1 | 4 | f(X) = \min_{1 \le i \le n} x_i | 1 | 7 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \min_{1 \le i \le n} x_i$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\cdot(\varepsilon))))$. $Gumbel(t)... | 0 | 4 | f(X) = \min_{1 \le i \le n} x_i | 1 | 8 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \frac{1}{n} \sum_{i=1}^n x_i$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\frac{1}{n}/(\varepsilon)))$. $Gum... | 1 | 5 | f(X) = \frac{1}{n} \sum_{i=1}^n x_i | \frac{1}{n} | 9 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \frac{1}{n} \sum_{i=1}^n x_i$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\frac{1}{n}/(2\cdot(\varepsilon)))... | 0 | 5 | f(X) = \frac{1}{n} \sum_{i=1}^n x_i | \frac{1}{n} | 10 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sqrt{\sum_{i=1}^n x_i^2}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\varepsilon)))$. $Gumbel(t)$ stand... | 1 | 6 | f(X) = \sqrt{\sum_{i=1}^n x_i^2} | 1 | 11 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sqrt{\sum_{i=1}^n x_i^2}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\cdot(\varepsilon))))$. $Gumbel(t... | 0 | 6 | f(X) = \sqrt{\sum_{i=1}^n x_i^2} | 1 | 12 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^{n-1} |x_i - x_{i+1}|$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot2/(\varepsilon)))$. $Gumbel(t)... | 1 | 7 | f(X) = \sum_{i=1}^{n-1} |x_i - x_{i+1}| | 2 | 13 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^{n-1} |x_i - x_{i+1}|$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot2/(2\cdot(\varepsilon))))$. $G... | 0 | 7 | f(X) = \sum_{i=1}^{n-1} |x_i - x_{i+1}| | 2 | 14 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n |x_i - \bar{x}|, \quad \bar{x} = \frac{1}{n} \sum_{j=1}^n x_j$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gu... | 1 | 8 | f(X) = \sum_{i=1}^n |x_i - \bar{x}|, \quad \bar{x} = \frac{1}{n} \sum_{j=1}^n x_j | \frac{2(n-1)}{n} | 15 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n |x_i - \bar{x}|, \quad \bar{x} = \frac{1}{n} \sum_{j=1}^n x_j$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gu... | 0 | 8 | f(X) = \sum_{i=1}^n |x_i - \bar{x}|, \quad \bar{x} = \frac{1}{n} \sum_{j=1}^n x_j | \frac{2(n-1)}{n} | 16 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n (x_i - \bar{x})^2$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\frac{n-1}{n}/(\varepsilon)))$. ... | 1 | 9 | f(X) = \sum_{i=1}^n (x_i - \bar{x})^2 | \frac{n-1}{n} | 17 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n (x_i - \bar{x})^2$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\frac{n-1}{n}/(2\cdot(\varepsilo... | 0 | 9 | f(X) = \sum_{i=1}^n (x_i - \bar{x})^2 | \frac{n-1}{n} | 18 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n (x_i - \mu)^2, \quad \mu \in [0,1]$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\varepsilon)... | 1 | 10 | f(X) = \sum_{i=1}^n (x_i - \mu)^2, \quad \mu \in [0,1] | 1 | 19 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n (x_i - \mu)^2, \quad \mu \in [0,1]$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\cdot(\vare... | 0 | 10 | f(X) = \sum_{i=1}^n (x_i - \mu)^2, \quad \mu \in [0,1] | 1 | 20 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n x_i^2$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\varepsilon)))$. $Gumbel(t)$ stands for t... | 1 | 11 | f(X) = \sum_{i=1}^n x_i^2 | 1 | 21 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n x_i^2$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\cdot(\varepsilon))))$. $Gumbel(t)$ stan... | 0 | 11 | f(X) = \sum_{i=1}^n x_i^2 | 1 | 22 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n x_i^3$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\varepsilon)))$. $Gumbel(t)$ stands for t... | 1 | 12 | f(X) = \sum_{i=1}^n x_i^3 | 1 | 23 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n x_i^3$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\cdot(\varepsilon))))$. $Gumbel(t)$ stan... | 0 | 12 | f(X) = \sum_{i=1}^n x_i^3 | 1 | 24 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \sqrt{x_i}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\varepsilon)))$. $Gumbel(t)$ stands ... | 1 | 13 | f(X) = \sum_{i=1}^n \sqrt{x_i} | 1 | 25 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \sqrt{x_i}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\cdot(\varepsilon))))$. $Gumbel(t)$... | 0 | 13 | f(X) = \sum_{i=1}^n \sqrt{x_i} | 1 | 26 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \log(x_i + 1)$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\log(2)/(\varepsilon)))$. $Gumbel(t)... | 1 | 14 | f(X) = \sum_{i=1}^n \log(x_i + 1) | \log(2) | 27 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \log(x_i + 1)$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\log(2)/(2\cdot(\varepsilon))))$. $G... | 0 | 14 | f(X) = \sum_{i=1}^n \log(x_i + 1) | \log(2) | 28 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n e^{x_i}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdote-1/(\varepsilon)))$. $Gumbel(t)$ stands f... | 1 | 15 | f(X) = \sum_{i=1}^n e^{x_i} | e-1 | 29 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n e^{x_i}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdote-1/(2\cdot(\varepsilon))))$. $Gumbel(t)$ ... | 0 | 15 | f(X) = \sum_{i=1}^n e^{x_i} | e-1 | 30 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \sin(\pi x_i)$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\varepsilon)))$. $Gumbel(t)$ stan... | 1 | 16 | f(X) = \sum_{i=1}^n \sin(\pi x_i) | 1 | 31 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \sin(\pi x_i)$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\cdot(\varepsilon))))$. $Gumbel(... | 0 | 16 | f(X) = \sum_{i=1}^n \sin(\pi x_i) | 1 | 32 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \cos(\pi x_i)$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot2/(\varepsilon)))$. $Gumbel(t)$ stan... | 1 | 17 | f(X) = \sum_{i=1}^n \cos(\pi x_i) | 2 | 33 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \cos(\pi x_i)$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot2/(2\cdot(\varepsilon))))$. $Gumbel(... | 0 | 17 | f(X) = \sum_{i=1}^n \cos(\pi x_i) | 2 | 34 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \tan^{-1}(x_i)$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\frac{\pi}{4}/(\varepsilon)))$. $Gu... | 1 | 18 | f(X) = \sum_{i=1}^n \tan^{-1}(x_i) | \frac{\pi}{4} | 35 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \tan^{-1}(x_i)$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\frac{\pi}{4}/(2\cdot(\varepsilon))... | 0 | 18 | f(X) = \sum_{i=1}^n \tan^{-1}(x_i) | \frac{\pi}{4} | 36 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n x_i (1 - x_i)$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\frac{1}{4}/(\varepsilon)))$. $Gumbe... | 1 | 19 | f(X) = \sum_{i=1}^n x_i (1 - x_i) | \frac{1}{4} | 37 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n x_i (1 - x_i)$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\frac{1}{4}/(2\cdot(\varepsilon))))$... | 0 | 19 | f(X) = \sum_{i=1}^n x_i (1 - x_i) | \frac{1}{4} | 38 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n x_i^2 (1 - x_i)^2$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\frac{1}{16}/(\varepsilon)))$. $... | 1 | 20 | f(X) = \sum_{i=1}^n x_i^2 (1 - x_i)^2 | \frac{1}{16} | 39 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n x_i^2 (1 - x_i)^2$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\frac{1}{16}/(2\cdot(\varepsilon... | 0 | 20 | f(X) = \sum_{i=1}^n x_i^2 (1 - x_i)^2 | \frac{1}{16} | 40 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \text{median}(x_1, \dots, x_n)$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\varepsilon)))$. $Gumbel(t)$ ... | 1 | 21 | f(X) = \text{median}(x_1, \dots, x_n) | 1 | 41 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \text{median}(x_1, \dots, x_n)$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\cdot(\varepsilon))))$. $Gum... | 0 | 21 | f(X) = \text{median}(x_1, \dots, x_n) | 1 | 42 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = Q_3 - Q_1 \quad \text{(interquartile range)}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\varepsilon)))... | 1 | 22 | f(X) = Q_3 - Q_1 \quad \text{(interquartile range)} | 1 | 43 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = Q_3 - Q_1 \quad \text{(interquartile range)}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\cdot(\vareps... | 0 | 22 | f(X) = Q_3 - Q_1 \quad \text{(interquartile range)} | 1 | 44 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \#\{ i : x_i > 0.5 \}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\varepsilon)))$. $Gumbel(t)$ stands fo... | 1 | 23 | f(X) = \#\{ i : x_i > 0.5 \} | 1 | 45 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \#\{ i : x_i > 0.5 \}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\cdot(\varepsilon))))$. $Gumbel(t)$ s... | 0 | 23 | f(X) = \#\{ i : x_i > 0.5 \} | 1 | 46 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \#\{ i : x_i \in [a, b] \}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\varepsilon)))$. $Gumbel(t)$ stan... | 1 | 24 | f(X) = \#\{ i : x_i \in [a, b] \} | 1 | 47 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \#\{ i : x_i \in [a, b] \}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\cdot(\varepsilon))))$. $Gumbel(... | 0 | 24 | f(X) = \#\{ i : x_i \in [a, b] \} | 1 | 48 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^{n-1} x_i x_{i+1}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot2/(\varepsilon)))$. $Gumbel(t)$ st... | 1 | 25 | f(X) = \sum_{i=1}^{n-1} x_i x_{i+1} | 2 | 49 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^{n-1} x_i x_{i+1}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot2/(2\cdot(\varepsilon))))$. $Gumbe... | 0 | 25 | f(X) = \sum_{i=1}^{n-1} x_i x_{i+1} | 2 | 50 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{1 \le i < j \le n} x_i x_j$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot(n-1)/(\varepsilon)))$. $Gumbe... | 1 | 26 | f(X) = \sum_{1 \le i < j \le n} x_i x_j | (n-1) | 51 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{1 \le i < j \le n} x_i x_j$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot(n-1)/(2\cdot(\varepsilon))))$... | 0 | 26 | f(X) = \sum_{1 \le i < j \le n} x_i x_j | (n-1) | 52 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^{\lfloor n/2 \rfloor} (x_{2i-1} - x_{2i})$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\vareps... | 1 | 27 | f(X) = \sum_{i=1}^{\lfloor n/2 \rfloor} (x_{2i-1} - x_{2i}) | 1 | 53 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^{\lfloor n/2 \rfloor} (x_{2i-1} - x_{2i})$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\cdot(... | 0 | 27 | f(X) = \sum_{i=1}^{\lfloor n/2 \rfloor} (x_{2i-1} - x_{2i}) | 1 | 54 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \left( \sum_{i=1}^n x_i \right)^2$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot(2n-1)/(\varepsilon)))$. $Gum... | 1 | 28 | f(X) = \left( \sum_{i=1}^n x_i \right)^2 | (2n-1) | 55 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \left( \sum_{i=1}^n x_i \right)^2$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot(2n-1)/(2\cdot(\varepsilon)))... | 0 | 28 | f(X) = \left( \sum_{i=1}^n x_i \right)^2 | (2n-1) | 56 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n i x_i$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdotn/(\varepsilon)))$. $Gumbel(t)$ stands for t... | 1 | 29 | f(X) = \sum_{i=1}^n i x_i | n | 57 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n i x_i$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdotn/(2\cdot(\varepsilon))))$. $Gumbel(t)$ stan... | 0 | 29 | f(X) = \sum_{i=1}^n i x_i | n | 58 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \mathbb{1} \left[ \sum_{i=1}^n x_i > \frac{n}{2} \right]$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\va... | 1 | 30 | f(X) = \mathbb{1} \left[ \sum_{i=1}^n x_i > \frac{n}{2} \right] | 1 | 59 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \mathbb{1} \left[ \sum_{i=1}^n x_i > \frac{n}{2} \right]$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\c... | 0 | 30 | f(X) = \mathbb{1} \left[ \sum_{i=1}^n x_i > \frac{n}{2} \right] | 1 | 60 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \mathbb{1}[x_i > 0.7]$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\varepsilon)))$. $Gumbel(... | 1 | 31 | f(X) = \sum_{i=1}^n \mathbb{1}[x_i > 0.7] | 1 | 61 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \mathbb{1}[x_i > 0.7]$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\cdot(\varepsilon))))$. ... | 0 | 31 | f(X) = \sum_{i=1}^n \mathbb{1}[x_i > 0.7] | 1 | 62 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \prod_{i=1}^n \mathbb{1}[x_i > \tau]$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\varepsilon)))$. $Gumbe... | 1 | 32 | f(X) = \prod_{i=1}^n \mathbb{1}[x_i > \tau] | 1 | 63 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \prod_{i=1}^n \mathbb{1}[x_i > \tau]$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\cdot(\varepsilon))))$... | 0 | 32 | f(X) = \prod_{i=1}^n \mathbb{1}[x_i > \tau] | 1 | 64 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \mathbb{1}[x_1 > x_2]$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\varepsilon)))$. $Gumbel(t)$ stands fo... | 1 | 33 | f(X) = \mathbb{1}[x_1 > x_2] | 1 | 65 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \mathbb{1}[x_1 > x_2]$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\cdot(\varepsilon))))$. $Gumbel(t)$ s... | 0 | 33 | f(X) = \mathbb{1}[x_1 > x_2] | 1 | 66 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \mathbb{1}[x_i > 0.5] \bmod 2$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\varepsilon)))$. ... | 1 | 34 | f(X) = \sum_{i=1}^n \mathbb{1}[x_i > 0.5] \bmod 2 | 1 | 67 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \mathbb{1}[x_i > 0.5] \bmod 2$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\cdot(\varepsilo... | 0 | 34 | f(X) = \sum_{i=1}^n \mathbb{1}[x_i > 0.5] \bmod 2 | 1 | 68 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \log\left(1 + \sum_{i=1}^n x_i^2\right)$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\log(2)/(\varepsilon)))... | 1 | 35 | f(X) = \log\left(1 + \sum_{i=1}^n x_i^2\right) | \log(2) | 69 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \log\left(1 + \sum_{i=1}^n x_i^2\right)$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\log(2)/(2\cdot(\vareps... | 0 | 35 | f(X) = \log\left(1 + \sum_{i=1}^n x_i^2\right) | \log(2) | 70 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sqrt{\sum_{i=1}^n (x_i - x_{i+1 \bmod n})^2}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\sqrt{2}/(\vareps... | 1 | 36 | f(X) = \sqrt{\sum_{i=1}^n (x_i - x_{i+1 \bmod n})^2} | \sqrt{2} | 71 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sqrt{\sum_{i=1}^n (x_i - x_{i+1 \bmod n})^2}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\sqrt{2}/(2\cdot(... | 0 | 36 | f(X) = \sqrt{\sum_{i=1}^n (x_i - x_{i+1 \bmod n})^2} | \sqrt{2} | 72 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \frac{\sum_{i=1}^n x_i^2}{1 + \sum_{i=1}^n x_i}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\frac{1}{2}/(\v... | 1 | 37 | f(X) = \frac{\sum_{i=1}^n x_i^2}{1 + \sum_{i=1}^n x_i} | \frac{1}{2} | 73 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \frac{\sum_{i=1}^n x_i^2}{1 + \sum_{i=1}^n x_i}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\frac{1}{2}/(2\... | 0 | 37 | f(X) = \frac{\sum_{i=1}^n x_i^2}{1 + \sum_{i=1}^n x_i} | \frac{1}{2} | 74 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \exp\left(-\sum_{i=1}^n x_i^2\right)$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot(1-e^{-1})/(\varepsilon)))... | 1 | 38 | f(X) = \exp\left(-\sum_{i=1}^n x_i^2\right) | (1-e^{-1}) | 75 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \exp\left(-\sum_{i=1}^n x_i^2\right)$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot(1-e^{-1})/(2\cdot(\vareps... | 0 | 38 | f(X) = \exp\left(-\sum_{i=1}^n x_i^2\right) | (1-e^{-1}) | 76 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \frac{1}{1 + x_i^2}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\frac{1}{2}/(\varepsilon)))$. ... | 1 | 39 | f(X) = \sum_{i=1}^n \frac{1}{1 + x_i^2} | \frac{1}{2} | 77 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \frac{1}{1 + x_i^2}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\frac{1}{2}/(2\cdot(\varepsilo... | 0 | 39 | f(X) = \sum_{i=1}^n \frac{1}{1 + x_i^2} | \frac{1}{2} | 78 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \sigma(x_i), \quad \sigma(x) = \frac{1}{1 + e^{-x}}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cd... | 1 | 40 | f(X) = \sum_{i=1}^n \sigma(x_i), \quad \sigma(x) = \frac{1}{1 + e^{-x}} | \frac{1-e^{-1}}{2(1 + e^{-1})} | 79 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \sigma(x_i), \quad \sigma(x) = \frac{1}{1 + e^{-x}}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cd... | 0 | 40 | f(X) = \sum_{i=1}^n \sigma(x_i), \quad \sigma(x) = \frac{1}{1 + e^{-x}} | \frac{1-e^{-1}}{2(1 + e^{-1})} | 80 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \frac{e^{x_i}}{\sum_{j=1}^n e^{x_j}} x_i$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\frac{e}{... | 1 | 41 | f(X) = \sum_{i=1}^n \frac{e^{x_i}}{\sum_{j=1}^n e^{x_j}} x_i | \frac{e}{e+n-1} | 81 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \frac{e^{x_i}}{\sum_{j=1}^n e^{x_j}} x_i$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\frac{e}{... | 0 | 41 | f(X) = \sum_{i=1}^n \frac{e^{x_i}}{\sum_{j=1}^n e^{x_j}} x_i | \frac{e}{e+n-1} | 82 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \left| x_i - \frac{1}{2} \right|$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\frac{1}{2}/(\var... | 1 | 42 | f(X) = \sum_{i=1}^n \left| x_i - \frac{1}{2} \right| | \frac{1}{2} | 83 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \left| x_i - \frac{1}{2} \right|$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\frac{1}{2}/(2\cd... | 0 | 42 | f(X) = \sum_{i=1}^n \left| x_i - \frac{1}{2} \right| | \frac{1}{2} | 84 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^{n-1} (x_i - x_{i+1})^2$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot2/(\varepsilon)))$. $Gumbel(... | 1 | 43 | f(X) = \sum_{i=1}^{n-1} (x_i - x_{i+1})^2 | 2 | 85 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^{n-1} (x_i - x_{i+1})^2$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot2/(2\cdot(\varepsilon))))$. ... | 0 | 43 | f(X) = \sum_{i=1}^{n-1} (x_i - x_{i+1})^2 | 2 | 86 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n x_{\pi(i)} \quad \text{for fixed permutation } \pi$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdo... | 1 | 44 | f(X) = \sum_{i=1}^n x_{\pi(i)} \quad \text{for fixed permutation } \pi | 1 | 87 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n x_{\pi(i)} \quad \text{for fixed permutation } \pi$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdo... | 0 | 44 | f(X) = \sum_{i=1}^n x_{\pi(i)} \quad \text{for fixed permutation } \pi | 1 | 88 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \text{rank of } x_k$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdotn-1/(\varepsilon)))$. $Gumbel(t)$ stands fo... | 1 | 45 | f(X) = \text{rank of } x_k | n-1 | 89 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \text{rank of } x_k$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdotn-1/(2\cdot(\varepsilon))))$. $Gumbel(t)$ s... | 0 | 45 | f(X) = \text{rank of } x_k | n-1 | 90 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{\substack{i=1 \\ i \text{ even}}}^n x_i$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\varepsilon)))... | 1 | 46 | f(X) = \sum_{\substack{i=1 \\ i \text{ even}}}^n x_i | 1 | 91 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{\substack{i=1 \\ i \text{ even}}}^n x_i$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\cdot(\vareps... | 0 | 46 | f(X) = \sum_{\substack{i=1 \\ i \text{ even}}}^n x_i | 1 | 92 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \frac{x_i}{i}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\varepsilon)))$. $Gumbel(t)$ stan... | 1 | 47 | f(X) = \sum_{i=1}^n \frac{x_i}{i} | 1 | 93 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \frac{x_i}{i}$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\cdot(\varepsilon))))$. $Gumbel(... | 0 | 47 | f(X) = \sum_{i=1}^n \frac{x_i}{i} | 1 | 94 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = -\sum_{i=1}^n x_i \log(x_i + \delta), \quad \delta > 1$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\log(1+\... | 1 | 48 | f(X) = -\sum_{i=1}^n x_i \log(x_i + \delta), \quad \delta > 1 | \log(1+\delta) | 95 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = -\sum_{i=1}^n x_i \log(x_i + \delta), \quad \delta > 1$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot\log(1+\... | 0 | 48 | f(X) = -\sum_{i=1}^n x_i \log(x_i + \delta), \quad \delta > 1 | \log(1+\delta) | 96 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \left| x_i - \frac{i}{n} \right|$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(\varepsilon)))... | 1 | 49 | f(X) = \sum_{i=1}^n \left| x_i - \frac{i}{n} \right| | 1 | 97 |
Given $D \in [0, 1]^{n \times d}$, where each row stands for a user and each column stands for their preference over $d$ items. Define score function $f(X) = \sum_{i=1}^n \left| x_i - \frac{i}{n} \right|$, where $X=(x_1, \cdots, x_n)\in [0, 1]^d$. Sample $Z_1, Z_2, ..., Z_d$ i.i.d. from $Gumbel((2\cdot1/(2\cdot(\vareps... | 0 | 49 | f(X) = \sum_{i=1}^n \left| x_i - \frac{i}{n} \right| | 1 | 98 |
DPrivBench: Benchmarking LLMs’ Reasoning for Differential Privacy
DPrivBench is a benchmark for evaluating whether language models can correctly reason about and verify claimed differential privacy (DP) guarantees from natural-language/LaTeX-format problem statements.
This release contains evaluation data from seven benchmark configs, along with one auxiliary function bank:
- Category 1: 6 fundamental mechanism tracks, each with 98 questions.
- Category 2: 125 more advanced algorithm-level DP questions derived from literatures.
The data files are stored as <config_name>/test-*.parquet.
Example Usage
from datasets import load_dataset
repo = "erchiw/DPrivBench"
# Category 1
ds = load_dataset(repo, "cate_1_Laplace_pureDP", split="test")
# Category 2
ds = load_dataset(repo, "cate_2", split="test")
Evaluation code can be found in the GitHub Repository
Dataset Structure
| Config | Description |
|---|---|
cate_1_Laplace_pureDP |
Laplace mechanism under pure DP |
cate_1_Gaussian_GDP |
Gaussian mechanism under GDP |
cate_1_Gaussian_zCDP |
Gaussian mechanism under zCDP |
cate_1_ExpoMech_pureDP |
Exponential mechanism under pure DP |
cate_1_LaplaceRNM_pureDP |
Report Noisy Max with Laplace noise |
cate_1_PF_pureDP |
Permute-and-Flip under pure DP |
cate_1_function_bank |
Function bank for category 1 questions |
cate_2 |
algorithm-level DP questions |
Task Format
Each example is a yes/no verification problem asking whether a mechanism or algorithm satisfies a claimed privacy guarantee under the stated assumptions.
Label formats
For all benchmark configs, the label field is an integer:
1= yes0= no
Data Fields
Category 1 (cate_1_*)
| Field | Description |
|---|---|
question_id |
Unique identifier for the question instance. |
question |
Text of the yes/no verification question. |
label |
Ground-truth label: 1 for yes, 0 for no. |
function_id |
Identifier of the query function in the function bank. |
function |
Mathematical definition of the function used in the question. |
function_sens |
L1 sensitivity of the function under a replace-one neighboring relation, assuming inputs in [0,1]. |
Category 2 (cate_2)
| Field | Description |
|---|---|
question_id |
Unique identifier for the question instance. |
question_tex |
Full question statement in LaTeX format. |
label |
Ground-truth label: 1 for yes, 0 for no. |
citation |
Bibliographic citation(s) for the relevant paper(s). Multiple entries may be separated by ;. |
negative_mode |
Construction type of the example. "atom" denotes a base positive/negative question; other values indicate how a negative or counterexample-style question was derived. |
pdf_link |
URL or pointer to the referenced source document(s). |
publish_year |
Publication year of the primary references. |
related_question |
question_id of the related base question, when applicable. Missing values may appear as NaN. |
section_number |
Section or location in the source where the relevant claim appears. |
subject |
Coarse-grained subject area. |
topic |
Fine-grained topic label within the subject. |
comments |
Short proof sketch or rationale explaining the correctness of the label. |
Notes
- This release is test-only and is intended for evaluation rather than training.
- The
cate_1_function_bankconfig is auxiliary and contains the function bank used to construct the Category 1 questions. For Category 1, we assume input data in $[0,1]$ and adopt the replace-one neighboring relation. - In Category 2, the neighboring relation and input data range are specified case-by-case in each question.
Citation
If you use this dataset, please cite the DPrivBench paper.
@misc{dprivbenchauthors,
title={DPrivBench: Benchmarking LLMs' Reasoning for Differential Privacy},
author={Erchi Wang and Pengrun Huang and Eli Chien and Om Thakkar and Kamalika Chaudhuri and Yu-Xiang Wang and Ruihan Wu},
year={2026},
eprint={2604.15851},
archivePrefix={arXiv},
url={https://arxiv.org/abs/2604.15851},
}
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