# Plug-and-Play Methods Provably Converge Even with Improperly Trained Denoisers: Convergence by Architectural Design

arXiv: https://arxiv.org/abs/2609.33087 · HTML: https://mlsanity.com/paper/2609.33087/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 32 of its 59 references that are in the map.
- First posted on arXiv 27 Sep 2026.

## Builds on (most cited first, 30 of 32)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [Plug-and-Play ADMM for Image Restoration: Fixed Point Convergence and Applications](https://mlsanity.com/paper/1605.01710/) (2016)
- [Sorting out Lipschitz function approximation](https://mlsanity.com/paper/1811.05381/) (2018)
- [Plug-and-Play Methods Provably Converge with Properly Trained Denoisers](https://mlsanity.com/paper/1905.05406/) (2019)
- [Directional convergence and alignment in deep learning](https://mlsanity.com/paper/2006.06657/) (2020)
- [Deep Equilibrium Architectures for Inverse Problems in Imaging](https://mlsanity.com/paper/2102.07944/) (2021)
- [Gradient Step Denoiser for convergent Plug-and-Play](https://mlsanity.com/paper/2110.03220/) (2021)
- [Regularization by Denoising: Clarifications and New Interpretations](https://mlsanity.com/paper/1806.02296/) (2018)
- [PnP-Flow: Plug-and-Play Image Restoration with Flow Matching](https://mlsanity.com/paper/2410.02423/) (2024)
- [Proximal Denoiser for Convergent Plug-and-Play Optimization with Nonconvex Regularization](https://mlsanity.com/paper/2201.13256/) (2022)
- [An Online Plug-and-Play Algorithm for Regularized Image Reconstruction](https://mlsanity.com/paper/1809.04693/) (2018)
- [Learning Maximally Monotone Operators for Image Recovery](https://mlsanity.com/paper/2012.13247/) (2020)
- [Regularization by Denoising via Fixed-Point Projection (RED-PRO)](https://mlsanity.com/paper/2008.00226/) (2020)
- [Scalable Plug-and-Play ADMM with Convergence Guarantees](https://mlsanity.com/paper/2006.03224/) (2020)
- [A Neural-Network-Based Convex Regularizer for Inverse Problems](https://mlsanity.com/paper/2211.12461/) (2022)
- [Provable Convergence of Plug-and-Play Priors with MMSE denoisers](https://mlsanity.com/paper/2005.07685/) (2020)
- [A characterization of proximity operators](https://mlsanity.com/paper/1807.04014/) (2018)
- [What's in a Prior? Learned Proximal Networks for Inverse Problems](https://mlsanity.com/paper/2310.14344/) (2023)
- [Recovery Analysis for Plug-and-Play Priors using the Restricted Eigenvalue Condition](https://mlsanity.com/paper/2106.03668/) (2021)
- [Differentiable Forward Projector for X-ray Computed Tomography](https://mlsanity.com/paper/2307.05801/) (2023)
- [Plug-and-play ISTA converges with kernel denoisers](https://mlsanity.com/paper/2004.03145/) (2020)
- [On Plug-and-Play Regularization using Linear Denoisers](https://mlsanity.com/paper/2105.05177/) (2021)
- [Convexity in ReLU Neural Networks: beyond ICNNs?](https://mlsanity.com/paper/2501.03017/) (2025)
- [Inexact Proximal Gradient Methods for Non-convex and Non-smooth Optimization](https://mlsanity.com/paper/1612.06003/) (2016)
- [Efficient Inexact Proximal Gradient Algorithm for Nonconvex Problems](https://mlsanity.com/paper/1612.09069/) (2016)
- [Nonasymptotic Convergence Rates for Plug-and-Play Methods With MMSE Denoisers](https://mlsanity.com/paper/2510.27211/) (2025)
- [Characterizations of inexact proximal operators](https://mlsanity.com/paper/2602.02022/) (2026)

## Abstract (arXiv)

Plug-and-Play (PnP) methods replace proximal operators with learned denoisers, which produce state-of-the-art reconstruction quality, but sacrifice the variational interpretation and convergence guarantees of proximal methods. Learned Proximal Networks (LPNs) address this problem by designing the denoiser architecture so that it is exactly the proximal operator of a regularizer. In this work, we clarify the theoretical foundations of LPNs and extend the framework to a broader class of activation functions. We then study two practical mechanisms for controlling the learned prior: (i) averaging the LPN with the identity, a common heuristic in PnP methods, and (ii) directly scaling the implicit regularizer induced by the proximal operator. In particular, we characterize the regularizer induced by averaging and develop a convergent method for evaluating the scaled proximal operator.