Generalization bounds for score-based generative models: a synthetic proof

TL;DR

This paper derives minimax convergence rates for score-based generative models under the Wasserstein distance, showing neural network estimators can achieve near-optimal rates under certain smoothness and tail conditions.

Contribution

It provides a unified, synthetic proof of convergence rates for score-based models, handling arbitrary smoothness and shape constraints, with more concise and general analysis.

Findings

Neural network score estimators achieve rate n^{-(β+1)/(2β+d)}.

The analysis applies to both deterministic and stochastic samplers.

Supports target densities with compact support or subGaussian tails.

Abstract

We establish minimax convergence rates for score-based generative models (SGMs) under the $1$-Wasserstein distance. Assuming the target density $p^\star$ lies in a nonparametric $\beta$-smooth H\"older class with either compact support or subGaussian tails on $\mathbb{R}^d$, we prove that neural network-based score estimators trained via denoising score matching yield generative models achieving rate $n^{-(\beta+1)/(2\beta+d)}$ up to polylogarithmic factors. Our unified analysis handles arbitrary smoothness $\beta > 0$, supports both deterministic and stochastic samplers, and leverages shape constraints on $p^\star$ to induce regularity of the score. The resulting proofs are more concise, and grounded in generic stability of diffusions and standard approximation theory.