Optimal Regularization Under Uncertainty: Distributional Robustness and Convexity Constraints

2025-10-03

AI Summary

This paper addresses the ill-posedness of inverse problems and statistical estimation under simultaneous distributional uncertainty and structural constraints (e.g., convexity). Methodologically, it proposes a distributionally robust regularization framework that models distributional ambiguity via a Wasserstein-1 uncertainty set and leverages convex duality to reformulate the original nonconvex/nonsmooth optimization into a tractable convex program. Theoretically, it characterizes how the robustness radius governs the behavior of the induced regularizer, naturally yielding a smooth, convex regularizer with bounded Lipschitz constant. The framework preserves structural priors while significantly enhancing robustness to distributional shifts, enabling controllable trade-offs between memorization and prior uniformity, and thereby improving deployment reliability under uncertainty.

Technology Category

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

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Regularization is a central tool for addressing ill-posedness in inverse problems and statistical estimation, with the choice of a suitable penalty often determining the reliability and interpretability of downstream solutions. While recent work has characterized optimal regularizers for well-specified data distributions, practical deployments are often complicated by distributional uncertainty and the need to enforce structural constraints such as convexity. In this paper, we introduce a framework for distributionally robust optimal regularization, which identifies regularizers that remain effective under perturbations of the data distribution. Our approach leverages convex duality to reformulate the underlying distributionally robust optimization problem, eliminating the inner maximization and yielding formulations that are amenable to numerical computation. We show how the resulting robust regularizers interpolate between memorization of the training distribution and uniform priors, providing insights into their behavior as robustness parameters vary. For example, we show how certain ambiguity sets, such as those based on the Wasserstein-1 distance, naturally induce regularity in the optimal regularizer by promoting regularizers with smaller Lipschitz constants. We further investigate the setting where regularizers are required to be convex, formulating a convex program for their computation and illustrating their stability with respect to distributional shifts. Taken together, our results provide both theoretical and computational foundations for designing regularizers that are reliable under model uncertainty and structurally constrained for robust deployment.

Problem

• Designing robust regularizers under data distribution uncertainty
• Ensuring convexity constraints in regularization for structural stability
• Addressing distribution shifts via distributionally robust optimization frameworks

Innovation

• Distributionally robust regularization under data uncertainty
• Convex duality reformulation for computational tractability
• Convexity-constrained regularizers ensuring structural stability