Generalized Schr\"odinger Bridge Matching

TL;DR

This paper introduces Generalized Schr"odinger Bridge Matching (GSBM), an algorithm that extends distribution matching techniques to incorporate task-specific objectives, improving stability and scalability in applications like image transfer and navigation.

Contribution

We propose GSBM, a novel algorithm that generalizes existing Schr"odinger Bridge methods to include task-specific costs and demonstrates superior stability and scalability.

Findings

Better preservation of feasible transport maps during training

Enhanced stability and convergence in various tasks

Significant scalability improvements over prior methods

Abstract

Modern distribution matching algorithms for training diffusion or flow models directly prescribe the time evolution of the marginal distributions between two boundary distributions. In this work, we consider a generalized distribution matching setup, where these marginals are only implicitly described as a solution to some task-specific objective function. The problem setup, known as the Generalized Schr\"odinger Bridge (GSB), appears prevalently in many scientific areas both within and without machine learning. We propose Generalized Schr\"odinger Bridge Matching (GSBM), a new matching algorithm inspired by recent advances, generalizing them beyond kinetic energy minimization and to account for task-specific state costs. We show that such a generalization can be cast as solving conditional stochastic optimal control, for which efficient variational approximations can be used, and further debiased with the aid of path integral theory. Compared to prior methods for solving GSB problems, our GSBM algorithm better preserves a feasible transport map between the boundary distributions throughout training, thereby enabling stable convergence and significantly improved scalability. We empirically validate our claims on an extensive suite of experimental setups, including crowd navigation, opinion depolarization, LiDAR manifolds, and image domain transfer. Our work brings new algorithmic opportunities for training diffusion models enhanced with task-specific optimality structures. Code available at https://github.com/facebookresearch/generalized-schrodinger-bridge-matching