Accelerating Convergence of Score-Based Diffusion Models, Provably

TL;DR

This paper introduces training-free algorithms that significantly accelerate the convergence of score-based diffusion models, providing provable convergence rates for both deterministic and stochastic samplers without relying on distribution smoothness.

Contribution

The paper develops novel, training-free acceleration algorithms for diffusion samplers with proven convergence rates, extending theoretical understanding of diffusion model speedups.

Findings

Deterministic sampler achieves $O(1/T^2)$ convergence rate.

Stochastic sampler achieves $O(1/T)$ convergence rate.

Algorithms do not require distribution smoothness or log-concavity.

Abstract

Score-based diffusion models, while achieving remarkable empirical performance, often suffer from low sampling speed, due to extensive function evaluations needed during the sampling phase. Despite a flurry of recent activities towards speeding up diffusion generative modeling in practice, theoretical underpinnings for acceleration techniques remain severely limited. In this paper, we design novel training-free algorithms to accelerate popular deterministic (i.e., DDIM) and stochastic (i.e., DDPM) samplers. Our accelerated deterministic sampler converges at a rate $O(1/{T}^2)$ with $T$ the number of steps, improving upon the $O(1/T)$ rate for the DDIM sampler; and our accelerated stochastic sampler converges at a rate $O(1/T)$, outperforming the rate $O(1/\sqrt{T})$ for the DDPM sampler. The design of our algorithms leverages insights from higher-order approximation, and shares similar intuitions as popular high-order ODE solvers like the DPM-Solver-2. Our theory accommodates $\ell_2$-accurate score estimates, and does not require log-concavity or smoothness on the target distribution.