Failure of uniform laws of large numbers for subdifferentials and beyond

2025-11-20

AI Summary

This paper addresses the fundamental question of whether subdifferentials satisfy a uniform law of large numbers (ULLN). Under the natural assumptions proposed by Shapiro and Xu, the authors construct two explicit counterexamples: one involving random Lipschitz functions and another involving random convex functions with finitely many smooth pieces—rigorously proving that the corresponding subdifferential sequences violate the ULLN. This result refutes the universality of the ULLN in nonsmooth stochastic optimization and reveals the intrinsic obstruction posed by nonsmoothness to uniform convergence. Methodologically, the analysis integrates tools from convex analysis and probability theory, leveraging structural properties of random convex functions to characterize the oscillatory behavior of the subdifferential as a set-valued mapping. The work resolves a long-standing open problem and delineates the theoretical boundaries of nonsmooth stochastic optimization, thereby establishing critical limitations for asymptotic analysis of statistical learning problems involving nondifferentiable loss functions.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization

Application Category

• Economics, Online Markets and Human Computation: Sustainability of Web economics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

We provide counterexamples showing that uniform laws of large numbers do not hold for subdifferentials under natural assumptions. Our results apply to random Lipschitz functions and random convex functions with a finite number of smooth pieces. Consequently, they resolve the questions posed by Shapiro and Xu [J. Math. Anal. Appl., 325(2), 2007] in the negative and highlight the obstacles nonsmoothness poses to uniform results.

Problem

• Demonstrates uniform laws fail for subdifferentials under natural assumptions
• Resolves Shapiro and Xu's 2007 questions about uniform convergence negatively
• Highlights obstacles nonsmoothness creates for uniform statistical results

Innovation

• Counterexamples disprove uniform subdifferential laws
• Results apply to random Lipschitz and convex functions
• Nonsmoothness creates obstacles for uniform convergence