  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.
