A Moment-Matching Approach to Testable Learning and a New Characterization of Rademacher Complexity

TL;DR

This paper introduces a new moment-matching approach for testable learning, linking it tightly to Rademacher complexity, and improves sample complexity bounds for various concept classes.

Contribution

It develops a novel algorithmic framework for testable learning using moment matching, extending previous results and establishing a fundamental link to Rademacher complexity.

Findings

Efficient testable learners for classes with low-degree sandwiching polynomials.

Sample complexity characterized by Rademacher complexity, matching information-theoretic bounds.

Uniform convergence is necessary and sufficient for testable learning.

Abstract

A remarkable recent paper by Rubinfeld and Vasilyan (2022) initiated the study of \emph{testable learning}, where the goal is to replace hard-to-verify distributional assumptions (such as Gaussianity) with efficiently testable ones and to require that the learner succeed whenever the unknown distribution passes the corresponding test. In this model, they gave an efficient algorithm for learning halfspaces under testable assumptions that are provably satisfied by Gaussians. In this paper we give a powerful new approach for developing algorithms for testable learning using tools from moment matching and metric distances in probability. We obtain efficient testable learners for any concept class that admits low-degree \emph{sandwiching polynomials}, capturing most important examples for which we have ordinary agnostic learners. We recover the results of Rubinfeld and Vasilyan as a corollary of our techniques while achieving improved, near-optimal sample complexity bounds for a broad range of concept classes and distributions. Surprisingly, we show that the information-theoretic sample complexity of testable learning is tightly characterized by the Rademacher complexity of the concept class, one of the most well-studied measures in statistical learning theory. In particular, uniform convergence is necessary and sufficient for testable learning. This leads to a fundamental separation from (ordinary) distribution-specific agnostic learning, where uniform convergence is sufficient but not necessary.