Sublinear iterations can suffice even for DDPMs

TL;DR

This paper introduces a new randomized midpoint integrator for DDPMs, achieving sublinear complexity in sample generation, and demonstrates its theoretical advantages and practical effectiveness in image synthesis.

Contribution

It presents the first sublinear complexity bound for pure DDPM sampling using a novel randomized midpoint method and the shifted composition rule framework.

Findings

Sublinear $ ilde{O}(\sqrt{d})$ score evaluations suffice for convergence.

The proposed method outperforms traditional discretization approaches.

Experimental results confirm practical efficiency in image synthesis.

Abstract

SDE-based methods such as denoising diffusion probabilistic models (DDPMs) have shown remarkable success in real-world sample generation tasks. Prior analyses of DDPMs have been focused on the exponential Euler discretization, showing guarantees that generally depend at least linearly on the dimension or initial Fisher information. Inspired by works in log-concave sampling (Shen and Lee, 2019), we analyze an integrator -- the denoising diffusion randomized midpoint method (DDRaM) -- that leverages an additional randomized midpoint to better approximate the SDE. Using a recently-developed analytic framework called the "shifted composition rule", we show that this algorithm enjoys favorable discretization properties under appropriate smoothness assumptions, with sublinear $\widetilde{O}(\sqrt{d})$ score evaluations needed to ensure convergence. This is the first sublinear complexity bound for pure DDPM sampling -- prior works which obtained such bounds worked instead with ODE-based sampling and had to make modifications to the sampler which deviate from how they are used in practice. We also provide experimental validation of the advantages of our method, showing that it performs well in practice with pre-trained image synthesis models.