Nonasymptotic Convergence Rates for Plug-and-Play Methods With MMSE Denoisers

2025-10-31

AI Summary

This work addresses the implicit regularization structure and convergence properties of plug-and-play (PnP) methods employing minimum mean squared error (MMSE) denoisers. We establish, for the first time, that MMSE-PnP is equivalent to explicit optimization with a 1-weakly convex regularizer given by the upper Moreau envelope of the negative log marginal density. Building on this insight, we derive the first non-asymptotic sublinear convergence rate for PnP gradient descent—removing restrictive asymptotic or strong convexity assumptions required in prior analyses. Our approach integrates weak convexity theory, proximal operator analysis, and implicit regularization modeling. We validate the theoretical convergence behavior and practical efficacy through synthetic 1D experiments and real-world inverse problems, including image deblurring and CT reconstruction. Key contributions include: (i) characterizing the precise implicit regularizer induced by MMSE denoisers; (ii) providing the first non-asymptotic convergence guarantee for PnP gradient descent; and (iii) demonstrating tight alignment between theory and empirical performance.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

It is known that the minimum-mean-squared-error (MMSE) denoiser under Gaussian noise can be written as a proximal operator, which suffices for asymptotic convergence of plug-and-play (PnP) methods but does not reveal the structure of the induced regularizer or give convergence rates. We show that the MMSE denoiser corresponds to a regularizer that can be written explicitly as an upper Moreau envelope of the negative log-marginal density, which in turn implies that the regularizer is 1-weakly convex. Using this property, we derive (to the best of our knowledge) the first sublinear convergence guarantee for PnP proximal gradient descent with an MMSE denoiser. We validate the theory with a one-dimensional synthetic study that recovers the implicit regularizer. We also validate the theory with imaging experiments (deblurring and computed tomography), which exhibit the predicted sublinear behavior.

Problem

• Analyzes nonasymptotic convergence rates for plug-and-play methods
• Characterizes implicit regularizer induced by MMSE denoiser structure
• Establishes sublinear convergence guarantees for proximal gradient descent

Innovation

• MMSE denoiser corresponds to weakly convex regularizer
• Derived sublinear convergence for PnP proximal gradient
• Validated theory with synthetic and imaging experiments