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 $\rho_0$ 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.