# A relaxed proximal gradient descent algorithm for convergent plug-and-play with proximal denoiser

arXiv: https://arxiv.org/abs/2301.13731 · HTML: https://mlsanity.com/paper/2301.13731/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 29 references that are in the map.
- First posted on arXiv 31 Jan 2023.

## Builds on (most cited first, 16 of 16)

- [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)
- [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)
- [Computational Optimal Transport: Complexity by Accelerated Gradient Descent Is Better Than by Sinkhorn's Algorithm](https://mlsanity.com/paper/1802.04367/) (2018)
- [Gradient Step Denoiser for convergent Plug-and-Play](https://mlsanity.com/paper/2110.03220/) (2021)
- [Proximal Denoiser for Convergent Plug-and-Play Optimization with Nonconvex Regularization](https://mlsanity.com/paper/2201.13256/) (2022)
- [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)
- [Convolutional Proximal Neural Networks and Plug-and-Play Algorithms](https://mlsanity.com/paper/2011.02281/) (2020)
- [An optimal randomized incremental gradient method](https://mlsanity.com/paper/1507.02000/) (2015)
- [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)
- [Douglas-Rachford splitting and ADMM for nonconvex optimization: tight convergence results](https://mlsanity.com/paper/1709.05747/) (2017)
- [A Survey on Nonconvex Regularization Based Sparse and Low-Rank Recovery in Signal Processing, Statistics, and Machine Learning](https://mlsanity.com/paper/1808.05403/) (2018)
- [The Primal-Dual Hybrid Gradient Method for Semiconvex Splittings](https://mlsanity.com/paper/1407.1723/) (2014)

## Built on by (most cited first, 12 of 12)

- [Learning Weakly Convex Regularizers for Convergent Image-Reconstruction Algorithms](https://mlsanity.com/paper/2308.10542/) (2023)
- [Provably Convergent Plug-and-Play Quasi-Newton Methods](https://mlsanity.com/paper/2303.07271/) (2023)
- [Convergent Bregman Plug-and-Play Image Restoration for Poisson Inverse Problems](https://mlsanity.com/paper/2306.03466/) (2023)
- [Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisation](https://mlsanity.com/paper/2402.01052/) (2024)
- [Convergent plug-and-play with proximal denoiser and unconstrained regularization parameter](https://mlsanity.com/paper/2311.01216/) (2023)
- [Stability of Data-Dependent Ridge-Regularization for Inverse Problems](https://mlsanity.com/paper/2406.12289/) (2024)
- [Efficient Unrolled Networks for Large-Scale 3D Inverse Problems](https://mlsanity.com/paper/2601.02141/) (2026)
- [GSNR: Graph Smooth Null-Space Representation for Inverse Problems](https://mlsanity.com/paper/2602.20328/) (2026)
- [Gradient Step Plug-and-Play Model for Dental Cone-Beam CT Reconstruction](https://mlsanity.com/paper/2605.28124/) (2026)
- [Enhanced Denoising and Convergent Regularisation Using Tweedie Scaling](https://mlsanity.com/paper/2503.05956/) (2025)
- [PG-DPIR: An efficient plug-and-play method for high-count Poisson-Gaussian inverse problems](https://mlsanity.com/paper/2504.10375/) (2025)
- [Convergent Plug-and-Play Image Restoration with Annealed Noise Levels](https://mlsanity.com/paper/2609.32393/) (2026)

## Abstract (arXiv)

This paper presents a new convergent Plug-and-Play (PnP) algorithm. PnP methods are efficient iterative algorithms for solving image inverse problems formulated as the minimization of the sum of a data-fidelity term and a regularization term. PnP methods perform regularization by plugging a pre-trained denoiser in a proximal algorithm, such as Proximal Gradient Descent (PGD). To ensure convergence of PnP schemes, many works study specific parametrizations of deep denoisers. However, existing results require either unverifiable or suboptimal hypotheses on the denoiser, or assume restrictive conditions on the parameters of the inverse problem. Observing that these limitations can be due to the proximal algorithm in use, we study a relaxed version of the PGD algorithm for minimizing the sum of a convex function and a weakly convex one. When plugged with a relaxed proximal denoiser, we show that the proposed PnP-αPGD algorithm converges for a wider range of regularization parameters, thus allowing more accurate image restoration.