Wasserstein Distributionally Robust Online Learning

TL;DR

This paper introduces a new online learning framework that uses Wasserstein distributional robustness to improve decision-making under worst-case distributional shifts, addressing computational challenges with tailored algorithms.

Contribution

It formulates online distributionally robust learning as a saddle-point game and develops efficient algorithms for piecewise concave losses, bridging infinite-dimensional optimization and budget allocation.

Findings

Framework converges to a robust Nash equilibrium.

Proposed algorithms outperform state-of-the-art solvers.

Achieves significant computational speedups.

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.