Schr\"odinger bridge problem via empirical risk minimization

2026-02-09

AI Summary

This work addresses the Schrödinger bridge problem under the practical setting where only samples from the endpoint distributions are available. The authors reformulate the Schrödinger system in terms of a positive transform potential function that satisfies a nonlinear fixed-point equation. They propose a novel approach that directly learns this potential function from a given function class via empirical risk minimization, leveraging a stochastic control representation to generate samples from the bridge process. This method circumvents conventional reliance on Sinkhorn iterations and kernel-smoothed dual differentials. Under a sub-Gaussian assumption, the authors establish uniform convergence of the empirical risk and demonstrate the efficacy and scalability of their approach through numerical experiments.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Social networks and social learning
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Schrödinger bridge
• empirical risk minimization
• endpoint distributions
• sample-based estimation

Innovation

• Schrödinger bridge
• empirical risk minimization
• nonlinear fixed-point equation
• stochastic control
• sample-based inference