We study monotonicity testing of functions $f \colon \{0,1\}^d \to \{0,1\}$ using sample-based algorithms, which are only allowed to observe the value of $f$ on points drawn independently from the uniform distribution. A classic result by Bshouty-Tamon (J. ACM 1996) proved that monotone functions can be learned with $\exp(O(\min\{\frac{1}{\varepsilon}\sqrt{d},d\}))$ samples and it is not hard to show that this bound extends to testing. Prior to our work the only lower bound for this problem was $\Omega(\sqrt{\exp(d)/\varepsilon})$ in the small $\varepsilon$ parameter regime, when $\varepsilon = O(d^{-3/2})$, due to Goldreich-Goldwasser-Lehman-Ron-Samorodnitsky (Combinatorica 2000). Thus, the sample complexity of monotonicity testing was wide open for $\varepsilon \gg d^{-3/2}$. We resolve this question, obtaining a tight lower bound of $\exp(\Omega(\min\{\frac{1}{\varepsilon}\sqrt{d},d\}))$ for all $\varepsilon$ at most a sufficiently small constant. In fact, we prove a much more general result, showing that the sample complexity of $k$-monotonicity testing and learning for functions $f \colon \{0,1\}^d \to [r]$ is $\exp(\Theta(\min\{\frac{rk}{\varepsilon}\sqrt{d},d\}))$. For testing with one-sided error we show that the sample complexity is $\exp(\Theta(d))$. Beyond the hypercube, we prove nearly tight bounds (up to polylog factors of $d,k,r,1/\varepsilon$ in the exponent) of $\exp(\widetilde{\Theta}(\min\{\frac{rk}{\varepsilon}\sqrt{d},d\}))$ on the sample complexity of testing and learning measurable $k$-monotone functions $f \colon \mathbb{R}^d \to [r]$ under product distributions. Our upper bound improves upon the previous bound of $\exp(\widetilde{O}(\min\{\frac{k}{\varepsilon^2}\sqrt{d},d\}))$ by Harms-Yoshida (ICALP 2022) for Boolean functions ($r=2$).

翻译：我们研究函数 $f \colon \{0,1\}^d \to \{0,1\}$ 的单调性测试问题，采用基于样本的算法——该类算法仅允许观测在均匀分布下独立抽取点上的函数值。Bshouty-Tamon (J. ACM 1996) 的经典结果表明，单调函数可通过 $\exp(O(\min\{\frac{1}{\varepsilon}\sqrt{d},d\}))$ 个样本学习，且该界可扩展至测试问题。在此工作之前，该问题的唯一下界是 Goldreich-Goldwasser-Lehman-Ron-Samorodnitsky (Combinatorica 2000) 在 $\varepsilon = O(d^{-3/2})$ 的小参数情形下给出的 $\Omega(\sqrt{\exp(d)/\varepsilon})$。因此，当 $\varepsilon \gg d^{-3/2}$ 时，单调性测试的样本复杂度尚属空白。我们解决了这一问题，针对所有小于充分小常数的 $\varepsilon$，建立了紧下界 $\exp(\Omega(\min\{\frac{1}{\varepsilon}\sqrt{d},d\}))$。事实上，我们证明了更一般的结果：对于函数 $f \colon \{0,1\}^d \to [r]$，$k$-单调性测试与学习的样本复杂度为 $\exp(\Theta(\min\{\frac{rk}{\varepsilon}\sqrt{d},d\}))$。对于单侧误差测试，我们证明样本复杂度为 $\exp(\Theta(d))$。在超立方体之外，我们证明了在乘积分布下可测 $k$-单调函数 $f \colon \mathbb{R}^d \to [r]$ 的测试与学习的样本复杂度近优界（指数中忽略 $d,k,r,1/\varepsilon$ 的多对数因子）为 $\exp(\widetilde{\Theta}(\min\{\frac{rk}{\varepsilon}\sqrt{d},d\}))$。我们的上界优于 Harms-Yoshida (ICALP 2022) 针对布尔函数 ($r=2$) 的先前结果 $\exp(\widetilde{O}(\min\{\frac{k}{\varepsilon^2}\sqrt{d},d\}))$。