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In this work, we present new proofs of convergence for Plug-and-Play (PnP)algorithms. PnP methods are efficient iterative algorithms for solving imageinverse problems where regularization is performed by plugging a pre-traineddenoiser in a proximal algorithm, such as Proximal Gradient Descent (PGD) orDouglas-Rachford Splitting (DRS). Recent research has explored convergence byincorporating 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 inuse then translates to restrictive conditions on the regularization parameterof the inverse problem. This can severely degrade the restoration capacity ofthe algorithm. In this paper, we present two remedies for this limitation.First, we provide a novel convergence proof for PnP-DRS that does not imposeany restrictions on the regularization parameter. Second, we examine a relaxedversion of the PGD algorithm that converges across a broader range ofregularization parameters. Our experimental study, conducted on deblurring andsuper-resolution experiments, demonstrate that both of these solutions enhancethe accuracy of image restoration.

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