Optimal Transport Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes

TL;DR

This paper studies optimal transport based distributionally robust optimization problems, revealing structural properties that enable the development of efficient algorithms with complexity comparable to standard stochastic gradient methods.

Contribution

It provides new structural insights into DRO problems with convex transport costs, facilitating the design of efficient optimization algorithms.

Findings

Structural properties of the value function and optimal policy are characterized.

DRO problems exhibit strong convexity even when the original problem does not.

Algorithms achieve sample and iteration complexity similar to stochastic gradient descent.

Abstract

We consider optimal transport based distributionally robust optimization (DRO) problems with locally strongly convex transport cost functions and affine decision rules. Under conventional convexity assumptions on the underlying loss function, we obtain structural results about the value function, the optimal policy, and the worst-case optimal transport adversarial model. These results expose a rich structure embedded in the DRO problem (e.g. strong convexity even if the non-DRO problem was not strongly convex, a suitable scaling of the Lagrangian for the DRO constraint, etc. which are crucial for the design of efficient algorithms). As a consequence of these results, one can develop efficient optimization procedures which have the same sample and iteration complexity as a natural non-DRO benchmark algorithm such as stochastic gradient descent.