Generative Modeling with Continuous Flows: Sample Complexity of Flow Matching

TL;DR

This paper analyzes the sample complexity of flow matching in generative models, showing that neural networks can learn velocity fields with a sample complexity of O(ε^{-4}) to achieve a Wasserstein-2 distance of O(ε).

Contribution

First theoretical analysis of sample complexity for flow matching generative models without relying on empirical risk minimization.

Findings

Neural networks can learn velocity fields with O(ε^{-4}) samples.

Decomposition of estimation error into approximation, statistical, and optimization errors.

Provides techniques for handling each error component independently.

Abstract

Flow matching has recently emerged as a promising alternative to diffusion-based generative models, offering faster sampling and simpler training by learning continuous flows governed by ordinary differential equations. Despite growing empirical success, the theoretical understanding of flow matching remains limited, particularly in terms of sample complexity results. In this work, we provide the first analysis of the sample complexity for flow-matching based generative models without assuming access to the empirical risk minimizer (ERM) of the loss function for estimating the velocity field. Under standard assumptions on the loss function for velocity field estimation and boundedness of the data distribution, we show that a sufficiently expressive neural network can learn a velocity field such that with $\mathcal{O}(\epsilon^{-4})$ samples, such that the Wasserstein-2 distance between the learned and the true distribution is less than $\mathcal{O}(\epsilon)$. The key technical idea is to decompose the velocity field estimation error into neural-network approximation error, statistical error due to the finite sample size, and optimization error due to the finite number of optimization steps for estimating the velocity field. Each of these terms are then handled via techniques that may be of independent interest.