Approximation and Generalization Abilities of Score-based Neural Network Generative Models for Sub-Gaussian Distributions

2025-05-16

AI Summary

This work studies score function estimation for score-based generative models (SGMs) given only $n$ i.i.d. $d$-dimensional samples from an $alpha$-sub-Gaussian true distribution $P_0$. Using deep ReLU neural networks and optimizing under both score matching loss and mean squared error, we establish—*for the first time*—near-optimal generalization rates *without* requiring strong assumptions such as Lipschitz continuity of the score or a lower bound on the data density. Specifically, the mean squared error achieves $ ilde{O}(n^{-1})$, while the score matching loss converges to $ ilde{O}(n^{-1} t_0^{-d/2})$ at time steps $t_0 gtrsim alpha^2 n^{-2/d} log n$. Our theory justifies early stopping to attain nearly minimax-optimal rates. Moreover, by characterizing model capacity via Sobolev and Besov space regularity, we reveal an intrinsic interplay between network architecture and the smoothness of the underlying distribution.

Technology Category

Machine Learning: Deep Generative Models & AutoencodersComputer Vision: Generative Adversarial Networks (GANs) for VisionNatural Language Processing: Learning & Optimization for NLP

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsSocial Networks and Social Media: Generative AI / large language models and their impact on social systemsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This paper studies the approximation and generalization abilities of score-based neural network generative models (SGMs) in estimating an unknown distribution $P_0$ from $n$ i.i.d. observations in $d$ dimensions. Assuming merely that $P_0$ is $alpha$-sub-Gaussian, we prove that for any time step $t in [t_0, n^{O(1)}]$, where $t_0 geq O(alpha^2n^{-2/d}log n)$, there exists a deep ReLU neural network with width $leq O(log^3n)$ and depth $leq O(n^{3/d}log_2n)$ that can approximate the scores with $ ilde{O}(n^{-1})$ mean square error and achieve a nearly optimal rate of $ ilde{O}(n^{-1}t_0^{-d/2})$ for score estimation, as measured by the score matching loss. Our framework is universal and can be used to establish convergence rates for SGMs under milder assumptions than previous work. For example, assuming further that the target density function $p_0$ lies in Sobolev or Besov classes, with an appropriately early stopping strategy, we demonstrate that neural network-based SGMs can attain nearly minimax convergence rates up to logarithmic factors. Our analysis removes several crucial assumptions, such as Lipschitz continuity of the score function or a strictly positive lower bound on the target density.

Problem

• Estimating unknown sub-Gaussian distributions using score-based neural networks
• Achieving optimal convergence rates for score estimation with deep ReLU networks
• Removing key assumptions like Lipschitz continuity in score-based generative models

Innovation

• Deep ReLU neural network for score approximation
• Nearly optimal score matching loss rate
• Universal framework under mild assumptions