Nearly Optimal Bounds for Sample-Based Testing and Learning of $k$-Monotone Functions

TL;DR

The paper resolves key questions in sample-based monotonicity testing and learning by establishing nearly tight exponential bounds under the uniform distribution for hypercube domains and extending to measurable $k$-monotone functions under product measures on $\mathbb{R}^d$. It introduces a general lower-bound framework using Talagrand random DNFs to show testing requires $\exp\big(\Omega(\min\{\frac{rk}{\varepsilon}\sqrt{d},d\})\big)$ samples, while providing matching upper bounds for learning and testing (up to polylog factors in the exponent) and a distinct $\exp(\Theta(d))$ bound for one-sided error testing. For continuous product spaces, it achieves nearly tight bounds $\exp(\widetilde{\Theta}(\min\{\frac{rk}{\varepsilon}\sqrt{d},d\}))$, improving prior $1/\varepsilon^2$-dependent exponents to $1/\varepsilon$ in the regime $r\ge 2$. Central techniques include a downsampling reduction to hypergrids, Fourier concentration and Low-Degree algorithms for learning on grids, and a layered DNFs construction to drive lower bounds. The results close the gap between sample-based and query-based models for these monotonicity classes and provide a foundation for further exploration of $k$-monotone function testing and learning in high dimensions.

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 $Ω(\sqrt{\exp(d)/\varepsilon})$ 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 $\exp(Ω(\min\{\frac{1}{\varepsilon}\sqrt{d},d\}))$ 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 $\exp(Ω(\min\{\frac{rk}{\varepsilon}\sqrt{d},d\}))$. For testing with one-sided error we show that the sample complexity is $\exp(Θ(d))$. Beyond the hypercube, we prove nearly tight bounds (up to polylog factors of $d,k,r,1/\varepsilon$ in the exponent) of $\exp(\widetildeΘ(\min\{\frac{rk}{\varepsilon}\sqrt{d},d\}))$ 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 ($r=2$).

Figures (1)

• Figure 1: An illustration of the construction used in our proof of \ref{['thm:LB_2sided_samples_rk']}. The image represents the set of points in the hypercube $\{0,1\}^d$ with Hamming weight in the interval $[\frac{d}{2},\frac{d}{2}+\varepsilon\sqrt{d})$, increasing from bottom to top. The numbers on the left denote the Hamming weight of the points lying in the adjacent horizontal line. The $B_i$ blocks are the sets of points contained between two adjacent horizontal lines. Each orange shaded region within $B_i$ represents the set of points satisfied by a term $t^{i,j}$. The blue numbers represent the value that functions in the support of $\mathcal{D}_{\texttt{yes}}$ and $\mathcal{D}_{\texttt{no}}$ can take. We have used the notation "$r-1,2$" as shorthand for $r-2,r-1$.

Theorems & Definitions (52)

• Theorem 1.1: Testing Lower Bound
• Corollary 1.2: Learning Lower Bound
• Theorem 1.3: Learning Upper Bound for Hypercubes
• Corollary 1.4: Testing Upper Bound for Hypercubes
• Theorem 1.5: Testing with One-Sided Error
• Theorem 1.6: Learning Upper Bound for Product Spaces
• Corollary 1.7: Testing Upper Bound for Product Spaces
• proof : Proof of \ref{['thm:UB-hypercube']}
• Definition 2.1
• Definition 2.2: $k$-monotonicity