Sample complexity of Schr\"odinger potential estimation

TL;DR

This paper investigates the sample complexity of estimating Schr"odinger potentials in generative models, providing bounds on the divergence between true and estimated distributions, which advances understanding of model generalization.

Contribution

It derives non-asymptotic high-probability bounds on the KL-divergence for Schr"odinger potential estimation, highlighting the potential for fast convergence even with unbounded supports.

Findings

Excess KL-risk can decrease as fast as O(log^2 n / n) with sample size n.

Provides non-asymptotic bounds on the divergence between true and estimated distributions.

Shows robustness of the estimation method under unbounded support conditions.

Abstract

We address the problem of Schr\"odinger potential estimation, which plays a crucial role in modern generative modelling approaches based on Schr\"odinger bridges and stochastic optimal control for SDEs. Given a simple prior diffusion process, these methods search for a path between two given distributions $\rho_0$ and $\rho_T^*$ requiring minimal efforts. The optimal drift in this case can be expressed through a Schr\"odinger potential. In the present paper, we study generalization ability of an empirical Kullback-Leibler (KL) risk minimizer over a class of admissible log-potentials aimed at fitting the marginal distribution at time $T$. Under reasonable assumptions on the target distribution $\rho_T^*$ and the prior process, we derive a non-asymptotic high-probability upper bound on the KL-divergence between $\rho_T^*$ and the terminal density corresponding to the estimated log-potential. In particular, we show that the excess KL-risk may decrease as fast as $O(\log^2 n / n)$ when the sample size $n$ tends to infinity even if both $\rho_0$ and $\rho_T^*$ have unbounded supports.