Generative Modeling with Continuous Flows: Sample Complexity of Flow Matching

Abstract

Flow matching has emerged as a promising alternative to diffusion-based generative modeling, offering a continuous-time ordinary differential equation formulation with fast sampling and a simple regression-based training objective. Despite its empirical success, some theoretical analyses idealize the learning procedure by assuming that the velocity field is obtained by exact empirical risk minimization. This assumption does not reflect the practical finite-dataset setting, where the empirical flow-matching loss is only approximately minimized by a finite number of stochastic-gradient updates. We develop an empirical-risk-minimizer-free analysis for flow matching that explicitly accounts for both finite-sample statistical error and finite-time empirical optimization error. Our analysis considers a finite dataset of \(N\) flow-matching samples and a learned parameter \(θ_T\) obtained after \(T\) stochastic gradient updates on the empirical objective. Under regularity assumptions on the interpolation path, sub-exponential source and target distributions, approximation by a bounded-parameter smooth neural velocity class, empirical-loss optimization, and uniform flow stability, we prove a guarantee that holds with high probability over the finite training dataset and in expectation over the stochastic-gradient randomness. The resulting bound separates approximation error, statistical error from the finite dataset, tail error from unbounded source and target distributions, and empirical optimization error from approximate minimization of the finite-sample objective.