Wasserstein Distributionally Robust Online Learning

2026-02-23

AI Summary

This work addresses the challenge of robust decision-making in online learning under distributional uncertainty by modeling distributionally robust online learning as a stochastic dynamic game between a decision-maker and a worst-case adversary, where ambiguity is characterized via a Wasserstein ambiguity set. The study establishes the first convergence analysis framework for this setting, proving that the proposed algorithm converges to a robust Nash equilibrium. Furthermore, it reveals an equivalence between worst-case expected optimization and the classical budget allocation problem, enabling the design of an efficient customized algorithm tailored for piecewise-concave loss functions. Theoretical guarantees ensure convergence, while empirical results demonstrate that the method significantly outperforms general-purpose solvers such as Gurobi, effectively alleviating computational bottlenecks in online settings.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We study distributionally robust online learning, where a risk-averse learner updates decisions sequentially to guard against worst-case distributions drawn from a Wasserstein ambiguity set centered at past observations. While this paradigm is well understood in the offline setting through Wasserstein Distributionally Robust Optimization (DRO), its online extension poses significant challenges in both convergence and computation. In this paper, we address these challenges. First, we formulate the problem as an online saddle-point stochastic game between a decision maker and an adversary selecting worst-case distributions, and propose a general framework that converges to a robust Nash equilibrium coinciding with the solution of the corresponding offline Wasserstein DRO problem. Second, we address the main computational bottleneck, which is the repeated solution of worst-case expectation problems. For the important class of piecewise concave loss functions, we propose a tailored algorithm that exploits problem geometry to achieve substantial speedups over state-of-the-art solvers such as Gurobi. The key insight is a novel connection between the worst-case expectation problem, an inherently infinite-dimensional optimization problem, and a classical and tractable budget allocation problem, which is of independent interest.

Problem

• distributionally robust optimization
• online learning
• Wasserstein ambiguity set
• worst-case distribution
• computational bottleneck

Innovation

• Wasserstein distributionally robust optimization
• online learning
• saddle-point stochastic game
• worst-case expectation
• budget allocation