Nearly Optimal Bounds for Sample-Based Testing and Learning of $k$-Monotone Functions (2310.12375v2)

Abstract: 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 $\exp(\widetilde{O}(\min{\frac{1}{\varepsilon}\sqrt{d},d}))$ 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 nearly 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 $\exp(\Omega(\min{\frac{rk}{\varepsilon}\sqrt{d},d}))$. 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 ($r=2$).