Learning and Testing Convex Functions

TL;DR

This paper investigates the learnability and testability of real-valued convex functions over Gaussian space, providing algorithms and bounds under Lipschitz smoothness assumptions in high dimensions.

Contribution

It introduces the first proper agnostic learning algorithm for Lipschitz convex functions in Gaussian space with specific sample complexity bounds, and develops testers for convexity with matching and exponential sample complexities.

Findings

Agnostic proper learning algorithm with polynomial sample complexity in 1/ε.

Lower bounds in the CSQ model indicating inherent difficulty.

Tolerant and one-sided convexity testers with specified sample complexities.

Abstract

We consider the problems of \emph{learning} and \emph{testing} real-valued convex functions over Gaussian space. Despite the extensive study of function convexity across mathematics, statistics, and computer science, its learnability and testability have largely been examined only in discrete or restricted settings -- typically with respect to the Hamming distance, which is ill-suited for real-valued functions. In contrast, we study these problems in high dimensions under the standard Gaussian measure, assuming sample access to the function and a mild smoothness condition, namely Lipschitzness. A smoothness assumption is natural and, in fact, necessary even in one dimension: without it, convexity cannot be inferred from finitely many samples. As our main results, we give: - Learning Convex Functions: An agnostic proper learning algorithm for Lipschitz convex functions that achieves error $\varepsilon$ using $n^{O(1/\varepsilon^2)}$ samples, together with a complementary lower bound of $n^{\mathrm{poly}(1/\varepsilon)}$ samples in the \emph{correlational statistical query (CSQ)} model. - Testing Convex Functions: A tolerant (two-sided) tester for convexity of Lipschitz functions with the same sample complexity (as a corollary of our learning result), and a one-sided tester (which never rejects convex functions) using $O(\sqrt{n}/\varepsilon)^n$ samples.