Sample complexity of Schrödinger potential estimation
This provides theoretical guarantees for Schrödinger bridge methods in generative modeling, addressing generalization ability for unbounded distributions, but it is incremental as it builds on existing frameworks.
The paper tackles the problem of estimating Schrödinger potentials in generative modeling by deriving a non-asymptotic upper bound on the KL-divergence between the target distribution and the estimated terminal density, showing an excess KL-risk decrease of O(log^2 n / n) with sample size n.
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 $ρ_0$ and $ρ_T^*$ 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 $ρ_T^*$ and the prior process, we derive a non-asymptotic high-probability upper bound on the KL-divergence between $ρ_T^*$ 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 $ρ_0$ and $ρ_T^*$ have unbounded supports.