Schr\"odinger bridge problem via empirical risk minimization

TL;DR

This paper introduces a novel learning-based method for solving the Schr"odinger bridge problem using empirical risk minimization, offering an alternative to classical iterative approaches and demonstrating promising numerical results.

Contribution

It reformulates the Schr"odinger system into a fixed-point problem and estimates the potential via empirical risk minimization, providing theoretical guarantees and practical implementation.

Findings

Uniform concentration of empirical risk established under sub-Gaussian assumptions

The learned potential effectively generates samples via stochastic control

Numerical experiments demonstrate the approach's effectiveness

Abstract

We study the Schr\"odinger bridge problem when the endpoint distributions are available only through samples. Classical computational approaches estimate Schr\"odinger potentials via Sinkhorn iterations on empirical measures and then construct a time-inhomogeneous drift by differentiating a kernel-smoothed dual solution. In contrast, we propose a learning-theoretic route: we rewrite the Schr\"odinger system in terms of a single positive transformed potential that satisfies a nonlinear fixed-point equation and estimate this potential by empirical risk minimization over a function class. We establish uniform concentration of the empirical risk around its population counterpart under sub-Gaussian assumptions on the reference kernel and terminal density. We plug the learned potential into a stochastic control representation of the bridge to generate samples. We illustrate performance of the suggested approach with numerical experiments.