Taming Score-Based Denoisers in ADMM: A Convergent Plug-and-Play Framework

TL;DR

This paper introduces a convergent plug-and-play ADMM framework using a novel three-stage denoiser, addressing manifold mismatch and convergence issues in score-based generative models for inverse problems.

Contribution

It proposes the AC-DC denoiser integrated into ADMM, with theoretical convergence guarantees and improved solution quality in inverse problem applications.

Findings

Ensures high-probability fixed-point ball convergence with proper parameters.

Achieves convergence with an adaptive step size under relaxed conditions.

Demonstrates improved solution quality over baselines in experiments.

Abstract

While score-based generative models have emerged as powerful priors for solving inverse problems, directly integrating them into optimization algorithms such as ADMM remains nontrivial. Two central challenges arise: i) the mismatch between the noisy data manifolds used to train the score functions and the geometry of ADMM iterates, especially due to the influence of dual variables, and ii) the lack of convergence understanding when ADMM is equipped with score-based denoisers. To address the manifold mismatch issue, we propose ADMM plug-and-play (ADMM-PnP) with the AC-DC denoiser, a new framework that embeds a three-stage denoiser into ADMM: (1) auto-correction (AC) via additive Gaussian noise, (2) directional correction (DC) using conditional Langevin dynamics, and (3) score-based denoising. In terms of convergence, we establish two results: first, under proper denoiser parameters, each ADMM iteration is a weakly nonexpansive operator, ensuring high-probability fixed-point $\textit{ball convergence}$ using a constant step size; second, under more relaxed conditions, the AC-DC denoiser is a bounded denoiser, which leads to convergence under an adaptive step size schedule. Experiments on a range of inverse problems demonstrate that our method consistently improves solution quality over a variety of baselines.