Gradient Flow Sampler-based Distributionally Robust Optimization

2025-10-29

AI Summary

This work addresses the challenge of efficiently sampling worst-case distributions in distributionally robust optimization (DRO). Methodologically, it establishes a mathematically rigorous gradient flow framework governed by partial differential equations, integrating Wasserstein–Fisher–Rao and Stein variational gradient flow theories into DRO. Coupled with Markov chain Monte Carlo (MCMC) sampling, the approach formulates a continuous-time evolution model over the space of probability distributions and designs corresponding discretization algorithms for solving both Wasserstein and Sinkhorn DRO problems. Key contributions include: (i) the first unified interpretation of several classical DRO algorithms as instances of gradient flows, elucidating their convergence dynamics and theoretical limits; (ii) a novel, interpretable, and analyzable paradigm for distributional optimization; and (iii) empirical validation—via numerical experiments—of effective worst-case distribution generation, algorithmic reproducibility, and performance improvement, thereby enhancing both theoretical depth and computational tractability.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• Proposes PDE gradient flow framework for distributionally robust optimization problems
• Develops sampling algorithms for worst-case distributions using gradient flow theory
• Solves Wasserstein and Sinkhorn DRO problems with novel gradient flow methods

Innovation

• PDE gradient flow framework for robust optimization
• Sampling worst-case distributions via gradient flow
• Solving Wasserstein DRO with Fisher-Rao flows