Large-Scale Methods for Distributionally Robust Optimization

TL;DR

This paper introduces scalable algorithms for distributionally robust optimization with CVaR and $ ext{chi}^2$ divergence, achieving theoretical efficiency guarantees suitable for large datasets and outperforming traditional methods in experiments.

Contribution

It provides the first gradient evaluation complexity bounds for $ ext{chi}^2$ uncertainty sets and improved linear scaling for CVaR, with proven optimality and practical efficiency.

Findings

Algorithms require gradient evaluations independent of dataset size.

Experimental results show 9-36 times efficiency gains over full-batch methods.

Theoretical guarantees are validated on MNIST and ImageNet datasets.

Abstract

We propose and analyze algorithms for distributionally robust optimization of convex losses with conditional value at risk (CVaR) and $\chi^2$ divergence uncertainty sets. We prove that our algorithms require a number of gradient evaluations independent of training set size and number of parameters, making them suitable for large-scale applications. For $\chi^2$ uncertainty sets these are the first such guarantees in the literature, and for CVaR our guarantees scale linearly in the uncertainty level rather than quadratically as in previous work. We also provide lower bounds proving the worst-case optimality of our algorithms for CVaR and a penalized version of the $\chi^2$ problem. Our primary technical contributions are novel bounds on the bias of batch robust risk estimation and the variance of a multilevel Monte Carlo gradient estimator due to [Blanchet & Glynn, 2015]. Experiments on MNIST and ImageNet confirm the theoretical scaling of our algorithms, which are 9--36 times more efficient than full-batch methods.