Learning and Testing Convex Functions

2025-11-14

AI Summary

This work studies learning and testing of high-dimensional real-valued convex functions under the standard Gaussian measure, overcoming limitations of prior convexity analysis confined to discrete domains (e.g., Hamming space). For Lipschitz-continuous convex functions, we establish the first learnability theory in continuous high-dimensional Gaussian space. We design a robust, two-sided tolerant tester and prove a lower bound on sample complexity in the statistical query model. Our adversarial learning algorithm achieves ε-error with sample complexity n^{O(1/ε²)}. We further construct both a consistent tester and a one-sided tester. Compared to existing results in discrete settings, our framework significantly improves model applicability—by operating under the natural Gaussian measure— theoretical completeness—via tight learnability and testability guarantees—and algorithmic efficiency—through polynomial sample dependence on 1/ε. The work thus advances the foundations of convex function learning and property testing in continuous, high-dimensional probabilistic spaces.

Technology Category

• Machine Learning: Learning Theory
• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Learning convex functions under Gaussian distribution with smoothness constraints
• Testing convexity of Lipschitz functions using sample-based methods
• Establishing sample complexity bounds for convex function learning and testing

Innovation

• Learning Lipschitz convex functions under Gaussian measure
• Testing convexity with tolerant two-sided sample complexity
• Providing one-sided tester using exponential sample size