Sample complexity of Schr"odinger potential estimation

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 ρ0 and ρT^* 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 ρT^* and the prior process, we derive a non-asymptotic high-probability upper bound on the KL-divergence between ρT^* 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(log2 n / n) when the sample size n tends to infinity even if both ρ0 and ρT^* have unbounded supports.