From Score Learning to Discretized Sampling: An End-to-End Generalization Analysis of Diffusion Models

2026-07-25

AI Summary

This work addresses the lack of a systematic theoretical understanding of how finite-sample learning, neural network parameterization, and numerical discretization jointly affect generation quality in diffusion models. The authors develop a unified framework for convergence and generalization analysis, decomposing the overall generation error— for the first time—into four interpretable components: forward truncation error, backward discretization error, generalization error (accounting for both data finiteness and forward discretization), and optimization gap. Leveraging a ResNet-type score estimator and combining tools from numerical analysis of stochastic differential equations with total variation distance bounds, they quantitatively characterize the joint influence of training sample size, temporal grid density, and optimization accuracy on generation fidelity, thereby establishing end-to-end theoretical guarantees.

Abstract

Despite the empirical success of score-based diffusion models, a complete theoretical understanding of how finite-sample learning, network parameterization, and numerical discretization jointly dictate generative quality remains underdeveloped. Existing sampling analyses often evaluate the generative performance conditional on an oracle score or a pre-specified error threshold. In this work, we establish a unified convergence and generalization framework for score-based diffusion models parameterized by practical ResNet-type architectures. We analyze the generalization and convergence properties from the practical finite-sample, discrete-time learning problem of the score function to the ideal continuous-time, population-level objective. Based on the generalization result of the learning problem of score function, we analyze the sampling process induced by the learned score function and provide an end-to-end total variation distance estimate for the generated terminal distribution. This estimate explicitly decomposes the overall generative error into four interpretable components: the truncation error of the forward process, the reverse-time discretization error, the generalization error incorporating both finite data and forward-time discretization, and the training optimization gap. Our results quantitatively characterize how the training sample size, temporal discretization grids, and optimization accuracy jointly control the final fidelity of samples generated by diffusion models.

Problem

• score-based diffusion models
• generalization
• discretization
• sampling error
• end-to-end analysis

Innovation

• score-based diffusion models
• generalization analysis
• discretized sampling
• end-to-end error decomposition
• ResNet parameterization