Large-Scale Methods for Distributionally Robust Optimization

• URL: http://arxiv.org/abs/2010.05893v2
• Date: Fri, 11 Dec 2020 03:05:30 GMT
• Title: Large-Scale Methods for Distributionally Robust Optimization
• Authors: Daniel Levy, Yair Carmon, John C. Duchi and Aaron Sidford
• Abstract summary: We prove that our algorithms require a number of evaluations gradient independent of training set size and number of parameters. Experiments on MNIST and ImageNet confirm the theoretical scaling of our algorithms, which are 9--36 times more efficient than full-batch methods.
• Score: 53.98643772533416
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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