Gradient Flow Sampler-based Distributionally Robust Optimization

TL;DR

This paper introduces a PDE gradient flow framework for distributionally robust optimization, enabling practical sampling algorithms for worst-case distributions and providing new insights into existing DRO methods.

Contribution

It presents a mathematically principled gradient flow approach to DRO, connecting sampling, optimization dynamics, and existing methods with novel theoretical insights.

Findings

Framework unifies sampling and optimization in DRO.

New algorithms for Wasserstein and Sinkhorn DRO problems.

Numerical results support theoretical claims.

Abstract

We propose a mathematically principled PDE gradient flow framework for distributionally robust optimization (DRO). Exploiting the recent advances in the intersection of Markov Chain Monte Carlo sampling and gradient flow theory, we show that our theoretical framework can be implemented as practical algorithms for sampling from worst-case distributions and, consequently, DRO. While numerous previous works have proposed various reformulation techniques and iterative algorithms, we contribute a sound gradient flow view of the distributional optimization that can be used to construct new algorithms. As an example of applications, we solve a class of Wasserstein and Sinkhorn DRO problems using the recently-discovered Wasserstein Fisher-Rao and Stein variational gradient flows. Notably, we also show some simple reductions of our framework recover exactly previously proposed popular DRO methods, and provide new insights into their theoretical limit and optimization dynamics. Numerical studies based on stochastic gradient descent provide empirical backing for our theoretical findings.