Iterative Sampling Methods for Sinkhorn Distributionally Robust Optimization

TL;DR

This paper introduces a primal reformulation of Sinkhorn distributionally robust optimization as a bilevel program, enabling simultaneous computation of robust decisions and worst-case distributions with provable algorithms.

Contribution

It presents a novel primal perspective on Sinkhorn DRO, reformulating it as a bilevel program and developing new sampling algorithms with theoretical guarantees.

Findings

Effective algorithms for Sinkhorn DRO with theoretical guarantees

Simultaneous computation of robust decision and worst-case distribution

Numerical validation on adversarial classification

Abstract

Distributionally robust optimization (DRO) has emerged as a powerful paradigm for reliable decision-making under uncertainty. This paper focuses on DRO with ambiguity sets defined via the Sinkhorn discrepancy: an entropy-regularized Wasserstein distance, referred to as Sinkhorn DRO. Existing work primarily addresses Sinkhorn DRO from a dual perspective, leveraging its formulation as a conditional stochastic optimization problem, for which many stochastic gradient methods are applicable. However, the theoretical analyses of such methods often rely on the boundedness of the loss function, and it is indirect to obtain the worst-case distribution associated with Sinkhorn DRO. In contrast, we study Sinkhorn DRO from the primal perspective, by reformulating it as a bilevel program with several infinite-dimensional lower-level subproblems over probability space. This formulation enables us to simultaneously obtain the optimal robust decision and the worst-case distribution, which is valuable in practical settings, such as generating stress-test scenarios or designing robust learning algorithms. We propose both double-loop and single-loop sampling-based algorithms with theoretical guarantees to solve this bilevel program. Finally, we demonstrate the effectiveness of our approach through a numerical study on adversarial classification.