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

2025/07/07 by Stéphanovitch, Arthur, Aamari, Eddie, Levrard, Clément · 1 citation
#FOS: Mathematics #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2507.04794

Abstract

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

Citations

• Regularity of the score function in generative models
• On the minimax optimality of Flow Matching through the connection to kernel density estimation
• Minimax Optimality of the Probability Flow ODE for Diffusion Models
• Generalization error bound for denoising score matching under relaxed manifold assumption
• Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
• Nonparametric estimation of a factorizable density using diffusion models
• Smooth transport map via diffusion process
• Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions
• From optimal score matching to optimal sampling
• A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models
• Flow matching achieves almost minimax optimal convergence
• Optimal score estimation via empirical Bayes smoothing
• Error Bounds for Flow Matching Methods
• The probability flow ODE is provably fast
• Diffusion Models are Minimax Optimal Distribution Estimators
• Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions
• Improving Lipschitz-Constrained Neural Networks by Learning Activation Functions
• Convergence of score-based generative modeling for general data distributions
• Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions
• Adapted Wasserstein distance between the laws of SDEs
• Elucidating the Design Space of Diffusion-Based Generative Models
• Equivariant Diffusion for Molecule Generation in 3D
• Diffusion Models Beat GANs on Image Synthesis
• Score-Based Generative Modeling through Stochastic Differential Equations
• DiffWave: A Versatile Diffusion Model for Audio Synthesis
• Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks
• Nonparametric Density Estimation & Convergence Rates for GANs under Besov IPM Losses
• Regularisation of Neural Networks by Enforcing Lipschitz Continuity
• A Connection Between Score Matching and Denoising Autoencoders

Cited by

• Distribution estimation via Flow Matching with Lipschitz guarantees

Related

• Regularity of the score function in generative models
• Convergence of score-based generative modeling for general data distributions
• Algorithm- and Data-Dependent Generalization Bounds for Diffusion Models
• Approximation and Generalization Abilities of Score-based Neural Network Generative Models for Sub-Gaussian Distributions
• Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions