A1409
Title: Optimal score function estimation via empirical risk minimization
Authors: Thomas Bonis - Universite Gustave Eiffel (France)
Thanh Mai Pham Ngoc - Universite Sorbonne Paris Nord (France) [presenting]
Viet Chi Tran - Centre Inria Universite Lille (France)
Abstract: Score-based generative models (SGMs) rely on estimating score functions via empirical risk minimization, yet their statistical properties under practical implementations remain poorly understood. Score estimation from i.i.d. data is studied and it is shown that minimax optimal rates over Sobolev classes can be achieved using empirical risk minimization with suitable smoothness constraints and derivative penalization. The diffusion framework of SGMs is then analyzed, where score functions must be estimated across noise levels. Despite singular behavior near zero noise (especially for data supported on low-dimensional manifolds) it is proved that properly constrained estimators achieve optimal rates for density estimation in Wasserstein distance. The results rely on the smoothing properties of the Ornstein-Uhlenbeck semigroup and provide theoretical support for the good generalization performance of neural network-based SGMs through implicit regularization.
