Metropolis-Adjusted Diffusion Models

TL;DR

This paper introduces Metropolis-adjusted diffusion models that correct bias in sampling by employing accept-reject steps, improving sample quality in score-based diffusion models.

Contribution

It proposes the first exact Metropolis-Hastings correction method for diffusion models, along with an efficient approximation to enhance sampling accuracy.

Findings

Improved sample quality demonstrated on synthetic and image datasets.

Achieved consistent FID score improvements in image generation.

Introduced a novel two-coin Bernoulli factory algorithm for exact correction.

Abstract

Sampling from score-based diffusion models incurs bias due to both time discretisation and the approximation of the score function. A common strategy for reducing this bias is to apply corrector steps based on the unadjusted Langevin algorithm (ULA) at each noise level within a predictor-corrector framework. However, ULA is itself a biased sampler, as it discretises a continuous diffusion process. In this work, we consider adjusted Langevin correctors that employ Metropolis--Hastings (MH) or Barker's accept-reject steps to correct for this bias. Since the target density ratio typically required by MH-based algorithms is unavailable, we propose methods that instead utilise the score function to compute the correct acceptance probability. We introduce the first exact method for adjusting Langevin corrections in diffusion models, based on a two-coin Bernoulli factory algorithm. We also propose an efficient approximation based on Simpson's rule that achieves accuracy of order $5/2$ in the step size at near-zero marginal cost. We demonstrate that these procedures improve sample quality on both synthetic and image datasets, yielding consistent gains in Fr\'echet Inception Distance (FID) on the latter.