Sample complexity of Schr""odinger potential estimation

2025-06-03

AI Summary

This paper investigates the sample complexity of estimating the Schrödinger potential in Schrödinger bridge generative modeling: given a prior diffusion process, how many samples are minimally required to learn the log-potential function such that the terminal distribution of the generated path exactly matches a target distribution ρₜ*. Methodologically, it employs empirical KL divergence minimization to estimate the log-potential, integrating Schrödinger bridge theory with rigorous diffusion process analysis. The main contribution is the first non-asymptotic, high-probability upper bound on the KL estimation error. Under mild assumptions—including unbounded supports of initial and terminal distributions—it establishes an excess risk convergence rate of O(log²n/n), substantially improving upon existing asymptotic or slower-rate results. The bound is tight and provides provable generalization guarantees, thereby furnishing a rigorous theoretical foundation for stochastic optimal control–based generative models.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

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 $ ho_0$ and $ ho_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 $ ho_T^*$ and the prior process, we derive a non-asymptotic high-probability upper bound on the KL-divergence between $ ho_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 $ ho_0$ and $ ho_T^*$ have unbounded supports.

Problem

• Estimating Schrödinger potential for generative modeling
• Generalization of KL risk minimizer for log-potentials
• Non-asymptotic bounds on KL-divergence for target distributions

Innovation

• Estimates Schrödinger potential via KL risk
• Uses empirical KL minimizer for log-potentials
• Derives non-asymptotic bounds for KL-divergence