Tight Bounds for Schr\"odinger Potential Estimation in Unpaired Data Translation

TL;DR

This paper establishes tight bounds on the generalization of Schr"odinger potential estimation from unpaired data, leveraging stochastic control and Ornstein-Uhlenbeck processes, with promising theoretical and empirical results.

Contribution

It introduces a novel theoretical framework for bounding the generalization error in Schr"odinger potential estimation using Ornstein-Uhlenbeck processes and empirical risk minimization.

Findings

Achieves near-fast convergence rates under certain conditions.

Provides tight bounds on generalization error.

Demonstrates effectiveness through numerical experiments.

Abstract

Modern methods of generative modelling and unpaired data translation based on Schr\"odinger bridges and stochastic optimal control theory aim to transform an initial density to a target one in an optimal way. In the present paper, we assume that we only have access to i.i.d. samples from the initial and final distributions. This makes our setup suitable for both generative modelling and unpaired data translation. Relying on the stochastic optimal control approach, we choose an Ornstein-Uhlenbeck process as the reference one and estimate the corresponding Schr\"odinger potential. Introducing a risk function as the Kullback-Leibler divergence between couplings, we derive tight bounds on the generalization ability of an empirical risk minimizer over a class of Schr\"odinger potentials, including Gaussian mixtures. Thanks to the mixing properties of the Ornstein-Uhlenbeck process, we almost achieve fast rates of convergence, up to some logarithmic factors, in favourable scenarios. We also illustrate the performance of the suggested approach with numerical experiments.