# Convergent plug-and-play with proximal denoiser and unconstrained regularization parameter

arXiv: https://arxiv.org/abs/2311.01216 · HTML: https://mlsanity.com/paper/2311.01216/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 55 references that are in the map.
- First posted on arXiv 2 Nov 2023.

## Builds on (most cited first, 27 of 27)

- [Plug-and-Play Image Restoration with Deep Denoiser Prior](https://mlsanity.com/paper/2008.13751/) (2020)
- [Learning Deep CNN Denoiser Prior for Image Restoration](https://mlsanity.com/paper/1704.03264/) (2017)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [Plug-and-Play Methods Provably Converge with Properly Trained Denoisers](https://mlsanity.com/paper/1905.05406/) (2019)
- [Gradient Step Denoiser for convergent Plug-and-Play](https://mlsanity.com/paper/2110.03220/) (2021)
- [Learning Proximal Operators: Using Denoising Networks for Regularizing Inverse Imaging Problems](https://mlsanity.com/paper/1704.03488/) (2017)
- [Proximal Denoiser for Convergent Plug-and-Play Optimization with Nonconvex Regularization](https://mlsanity.com/paper/2201.13256/) (2022)
- [Plug-and-Play Priors for Bright Field Electron Tomography and Sparse Interpolation](https://mlsanity.com/paper/1512.07331/) (2015)
- [Plug-and-Play Algorithms for Large-scale Snapshot Compressive Imaging](https://mlsanity.com/paper/2003.13654/) (2020)
- [Plug and play methods for magnetic resonance imaging (long version)](https://mlsanity.com/paper/1903.08616/) (2019)
- [Learning Maximally Monotone Operators for Image Recovery](https://mlsanity.com/paper/2012.13247/) (2020)
- [Scalable Plug-and-Play ADMM with Convergence Guarantees](https://mlsanity.com/paper/2006.03224/) (2020)
- [Global Convergence of Block Coordinate Descent in Deep Learning](https://mlsanity.com/paper/1803.00225/) (2018)
- [First Order Methods beyond Convexity and Lipschitz Gradient Continuity with Applications to Quadratic Inverse Problems](https://mlsanity.com/paper/1706.06461/) (2017)
- [Convolutional Proximal Neural Networks and Plug-and-Play Algorithms](https://mlsanity.com/paper/2011.02281/) (2020)
- [A Three-Operator Splitting Scheme and its Optimization Applications](https://mlsanity.com/paper/1504.01032/) (2015)
- [An optimal randomized incremental gradient method](https://mlsanity.com/paper/1507.02000/) (2015)
- [Block Coordinate Regularization by Denoising](https://mlsanity.com/paper/1905.05113/) (2019)
- [A characterization of proximity operators](https://mlsanity.com/paper/1807.04014/) (2018)
- [iPiano: Inertial Proximal Algorithm for Non-Convex Optimization](https://mlsanity.com/paper/1404.4805/) (2014)
- [RandProx: Primal-Dual Optimization Algorithms with Randomized Proximal Updates](https://mlsanity.com/paper/2207.12891/) (2022)
- [Douglas-Rachford splitting and ADMM for nonconvex optimization: tight convergence results](https://mlsanity.com/paper/1709.05747/) (2017)
- [Douglas-Rachford splitting for nonconvex optimization with application to nonconvex feasibility problems](https://mlsanity.com/paper/1409.8444/) (2014)
- [A relaxed proximal gradient descent algorithm for convergent plug-and-play with proximal denoiser](https://mlsanity.com/paper/2301.13731/) (2023)
- [Dualize, Split, Randomize: Toward Fast Nonsmooth Optimization Algorithms](https://mlsanity.com/paper/2004.02635/) (2020)
- [Averaged Deep Denoisers for Image Regularization](https://mlsanity.com/paper/2207.07321/) (2022)

## Built on by (most cited first, 5 of 5)

- [Equivariant plug-and-play image reconstruction](https://mlsanity.com/paper/2312.01831/) (2023)
- [FiRe: Fixed-points of Restoration Priors for Solving Inverse Problems](https://mlsanity.com/paper/2411.18970/) (2024)
- [Learning Cocoercive Conservative Denoisers via Helmholtz Decomposition for Poisson Inverse Problems](https://mlsanity.com/paper/2505.08909/) (2025)
- [Nonasymptotic Convergence Rates for Plug-and-Play Methods With MMSE Denoisers](https://mlsanity.com/paper/2510.27211/) (2025)
- [Solving Imaging Inverse Problems Using Plug-and-Play Denoisers: Regularization and Optimization Perspectives](https://mlsanity.com/paper/2509.03475/) (2025)

## Abstract (arXiv)

In this work, we present new proofs of convergence for Plug-and-Play (PnP) algorithms. PnP methods are efficient iterative algorithms for solving image inverse problems where regularization is performed by plugging a pre-trained denoiser in a proximal algorithm, such as Proximal Gradient Descent (PGD) or Douglas-Rachford Splitting (DRS). Recent research has explored convergence by incorporating a denoiser that writes exactly as a proximal operator. However, the corresponding PnP algorithm has then to be run with stepsize equal to 1. The stepsize condition for nonconvex convergence of the proximal algorithm in use then translates to restrictive conditions on the regularization parameter of the inverse problem. This can severely degrade the restoration capacity of the algorithm. In this paper, we present two remedies for this limitation. First, we provide a novel convergence proof for PnP-DRS that does not impose any restrictions on the regularization parameter. Second, we examine a relaxed version of the PGD algorithm that converges across a broader range of regularization parameters. Our experimental study, conducted on deblurring and super-resolution experiments, demonstrate that both of these solutions enhance the accuracy of image restoration.