Downsampling for Testing and Learning in Product Distributions

TL;DR

This paper introduces a novel downsampling technique for distribution-free property testing and learning over product distributions in high-dimensional spaces, improving efficiency and broadening applicability.

Contribution

The authors develop a general downsampling method that leverages rectilinear isoperimetry, enabling new efficient algorithms for testing and learning in product distributions.

Findings

Improved monotonicity testing complexity from O(d^7) to ~O(d^3).

Polynomial-time agnostic learning for functions of a constant number of halfspaces.

Efficient algorithms for convex sets and k-alternating functions with exponential dependence on dimension.

Abstract

We study distribution-free property testing and learning problems where the unknown probability distribution is a product distribution over $\mathbb{R}^d$. For many important classes of functions, such as intersections of halfspaces, polynomial threshold functions, convex sets, and $k$-alternating functions, the known algorithms either have complexity that depends on the support size of the distribution, or are proven to work only for specific examples of product distributions. We introduce a general method, which we call downsampling, that resolves these issues. Downsampling uses a notion of "rectilinear isoperimetry" for product distributions, which further strengthens the connection between isoperimetry, testing, and learning. Using this technique, we attain new efficient distribution-free algorithms under product distributions on $\mathbb{R}^d$: 1. A simpler proof for non-adaptive, one-sided monotonicity testing of functions $[n]^d \to \{0,1\}$, and improved sample complexity for testing monotonicity over unknown product distributions, from $O(d^7)$ [Black, Chakrabarty, & Seshadhri, SODA 2020] to $\widetilde O(d^3)$. 2. Polynomial-time agnostic learning algorithms for functions of a constant number of halfspaces, and constant-degree polynomial threshold functions. 3. An $\exp(O(d \log(dk)))$-time agnostic learning algorithm, and an $\exp(O(d \log(dk)))$-sample tolerant tester, for functions of $k$ convex sets; and a $2^{\widetilde O(d)}$ sample-based one-sided tester for convex sets. 4. An $\exp(\widetilde O(k \sqrt d))$-time agnostic learning algorithm for $k$-alternating functions, and a sample-based tolerant tester with the same complexity.