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Generative Modeling with Continuous Flows: Sample Complexity of Flow Matching

Mudit Gaur, Prashant Trivedi, Shuchin Aeron, Amrit Singh Bedi, George K. Atia, Vaneet Aggarwal

TL;DR

This work analyzes the sample efficiency of flow-matching generative models by recasting them as deterministic ODE flows driven by a velocity field learned via neural networks. It introduces a Gaussian probability path to achieve a tractable, bounded-analysis setting and decomposes the velocity-estimation error into approximation, statistical, and optimization components. The authors prove a first formal sample complexity bound for flow matching without requiring ERM, showing that a fully expressive neural network can achieve $W_2$ distance $\mathcal{O}(\varepsilon)$ with $n = \Omega((W)^{2D-2} d^2 / \varepsilon^4 \log(2/\delta))$ samples, up to a model-approximation error. This work bridges theory and practice for flow-based generative modeling, providing foundations for data-efficient training and fast sampling in continuous-flow frameworks.

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 $\mathcal{O}(ε^{-4})$ samples, such that the Wasserstein-2 distance between the learned and the true distribution is less than $\mathcal{O}(ε)$. 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.

Generative Modeling with Continuous Flows: Sample Complexity of Flow Matching

TL;DR

This work analyzes the sample efficiency of flow-matching generative models by recasting them as deterministic ODE flows driven by a velocity field learned via neural networks. It introduces a Gaussian probability path to achieve a tractable, bounded-analysis setting and decomposes the velocity-estimation error into approximation, statistical, and optimization components. The authors prove a first formal sample complexity bound for flow matching without requiring ERM, showing that a fully expressive neural network can achieve distance with samples, up to a model-approximation error. This work bridges theory and practice for flow-based generative modeling, providing foundations for data-efficient training and fast sampling in continuous-flow frameworks.

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 samples, such that the Wasserstein-2 distance between the learned and the true distribution is less than . 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.

Paper Structure

This paper contains 17 sections, 10 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries and Problem Formulation
  4. Construction of a generative model via an ODE.
  5. A result from benton2024error.
  6. Wasserstein distance and its relation to velocity estimation.
  7. Problem statement.
  8. Our Approach
  9. Gaussian Probability Path.
  10. Theoretical Results
  11. Conclusion
  12. Proof of Lemma \ref{['lemma:statistical_error']}
  13. Proof of Lemma \ref{['lemma:opt_error']}
  14. Final Theoretical Result
  15. Intermediate Lemmas
  16. ...and 2 more sections

Key Result

Theorem 2.1

Suppose that $\pi_0, \pi_1$ are initial and target probability distributions respectively on $\mathbb{R}^d$ , $Y$ is the flow starting in $\pi_0$ with velocity field $u_\theta$, and $\hat{\pi}_1$ is the law of $Y_1$. Also, suppose the following assumptions hold. Under the above assumptions, we have

Theorems & Definitions (17)

  • Theorem 2.1: Theorem 1 benton2024error
  • Theorem 4.1
  • Lemma 4.1: Approximation Error
  • Lemma 4.2: Statistical Error
  • Lemma 4.3: Optimization Error
  • Proof A.1
  • Proof B.1
  • Proof C.1
  • Lemma D.1: Theorem 26.5 of shalev2014understanding
  • Lemma D.2: Extension of Massart's Lemma bousquet2003introduction
  • ...and 7 more