Sample complexity of Schrödinger potential estimation

Nikita Puchkin, Iurii Pustovalov, Yuri Sapronov, Denis Suchkov, Alexey Naumov, Denis Belomestny

Keywords: Schrödinger bridge, stochastic optimal control, Schrödinger potential, high-probability bounds, excess risk

TL;DR: Sharper risk bounds in the problem of Schrödinger potential estimation, which is connected with modern generative modelling approaches.

Abstract: We address the problem of Schrödinger potential estimation, which plays a crucial role in modern generative modelling approaches based on Schrödinger 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 Schrödinger 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 $\mathcal O(\log n / n)$ when the sample size $n$ tends to infinity even if both $\rho_0$ and $\rho_T$ have unbounded supports.

Primary Area: Theory (e.g., control theory, learning theory, algorithmic game theory)

Submission Number: 22589