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A New Convergence Analysis of Plug-and-Play Proximal Gradient Descent Under Prior Mismatch

Guixian Xu, Jinglai Li, Junqi Tang

TL;DR

This work delivers the first convergence proof for plug-and-play proximal gradient descent under prior mismatch, where the denoiser is trained on a different data distribution than the inference task. By recasting the denoiser as the proximal operator of a (potentially nonconvex) regularizer φ_σ and formulating the objective F_{λ,σ} = λ f + φ_σ, the authors derive descent guarantees under relaxed assumptions, accommodating nonconvex f and g_σ as well as expansive denoisers. The analysis introduces an error term ε_k capturing mismatches and shows that if ε_k decays summably, the gradient norm of F_{λ,σ} converges to zero; in the matched (target) denoiser case, a tighter bound is obtained without the ε_k term. These results provide theoretical guarantees for PnP-PGD in realistic mismatched-prior scenarios and offer guidance for tuning λ and σ in practice, broadening the applicability of plug-and-play methods to nonconvex and distribution-shifted settings.

Abstract

In this work, we provide a new convergence theory for plug-and-play proximal gradient descent (PnP-PGD) under prior mismatch where the denoiser is trained on a different data distribution to the inference task at hand. To the best of our knowledge, this is the first convergence proof of PnP-PGD under prior mismatch. Compared with the existing theoretical results for PnP algorithms, our new results removed the need for several restrictive and unverifiable assumptions.

A New Convergence Analysis of Plug-and-Play Proximal Gradient Descent Under Prior Mismatch

TL;DR

This work delivers the first convergence proof for plug-and-play proximal gradient descent under prior mismatch, where the denoiser is trained on a different data distribution than the inference task. By recasting the denoiser as the proximal operator of a (potentially nonconvex) regularizer φ_σ and formulating the objective F_{λ,σ} = λ f + φ_σ, the authors derive descent guarantees under relaxed assumptions, accommodating nonconvex f and g_σ as well as expansive denoisers. The analysis introduces an error term ε_k capturing mismatches and shows that if ε_k decays summably, the gradient norm of F_{λ,σ} converges to zero; in the matched (target) denoiser case, a tighter bound is obtained without the ε_k term. These results provide theoretical guarantees for PnP-PGD in realistic mismatched-prior scenarios and offer guidance for tuning λ and σ in practice, broadening the applicability of plug-and-play methods to nonconvex and distribution-shifted settings.

Abstract

In this work, we provide a new convergence theory for plug-and-play proximal gradient descent (PnP-PGD) under prior mismatch where the denoiser is trained on a different data distribution to the inference task at hand. To the best of our knowledge, this is the first convergence proof of PnP-PGD under prior mismatch. Compared with the existing theoretical results for PnP algorithms, our new results removed the need for several restrictive and unverifiable assumptions.
Paper Structure (9 sections, 7 theorems, 67 equations, 1 table)

This paper contains 9 sections, 7 theorems, 67 equations, 1 table.

Table of Contents

  1. Introduction
  2. Theory
  3. PnP-PGD with Mismatched Denoiser
  4. Theoretical Analysis
  5. Proofs
  6. Warm-up
  7. Proof of Lemma \ref{['lemma: error_shrink']}
  8. Proof of Theorem \ref{['them: 1']}
  9. Proof of Theorem \ref{['them: 2']}

Key Result

Proposition 2.1

Let $\mathcal{X}$ be an open convex subset of $\mathbb{R}^n$ and $g_\sigma: \mathcal{X} \to \mathbb{R}$ a $\mathcal{C}^{k+1}$ function with $k \geq 1$ and $\nabla g_\sigma$$L$-Lipschitz with $L < 1$. Then, for $h_\sigma : \boldsymbol{x} \to \frac{1}{2} \|\boldsymbol{x}\|^2 - g_\sigma(\boldsymbol{x})

Theorems & Definitions (8)

  • Proposition 2.1: hurault2022proximal
  • Remark 2.1
  • Theorem 2.1
  • Lemma 2.1
  • Theorem 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3