Wasserstein Convergence of Score-based Generative Models under Semiconvexity and Discontinuous Gradients

TL;DR

This paper provides the first non-asymptotic Wasserstein-2 convergence guarantees for score-based generative models targeting semiconvex distributions with potentially discontinuous gradients, broadening theoretical understanding.

Contribution

It establishes explicit, sharp convergence bounds for SGMs under semiconvexity without smoothness assumptions, applicable to complex, non-smooth data distributions.

Findings

Achieves optimal $O(\sqrt{d})$ dependence on data dimension

Provides convergence rate of order one

Applicable to a wide class of distributions including Gaussian mixtures and double-well potentials

Abstract

Score-based Generative Models (SGMs) approximate a data distribution by perturbing it with Gaussian noise and subsequently denoising it via a learned reverse diffusion process. These models excel at modeling complex data distributions and generating diverse samples, achieving state-of-the-art performance across domains such as computer vision, audio generation, reinforcement learning, and computational biology. Despite their empirical success, existing Wasserstein-2 convergence analysis typically assume strong regularity conditions-such as smoothness or strict log-concavity of the data distribution-that are rarely satisfied in practice. In this work, we establish the first non-asymptotic Wasserstein-2 convergence guarantees for SGMs targeting semiconvex distributions with potentially discontinuous gradients. Our upper bounds are explicit and sharp in key parameters, achieving optimal dependence of $O(\sqrt{d})$ on the data dimension $d$ and convergence rate of order one. The framework accommodates a wide class of practically relevant distributions, including symmetric modified half-normal distributions, Gaussian mixtures, double-well potentials, and elastic net potentials. By leveraging semiconvexity without requiring smoothness assumptions on the potential such as differentiability, our results substantially broaden the theoretical foundations of SGMs, bridging the gap between empirical success and rigorous guarantees in non-smooth, complex data regimes.