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This paper presents a new convergent Plug-and-Play (PnP) algorithm. PnPmethods are efficient iterative algorithms for solving image inverse problemsformulated as the minimization of the sum of a data-fidelity term and aregularization term. PnP methods perform regularization by plugging apre-trained denoiser in a proximal algorithm, such as Proximal Gradient Descent(PGD). To ensure convergence of PnP schemes, many works study specificparametrizations of deep denoisers. However, existing results require eitherunverifiable or suboptimal hypotheses on the denoiser, or assume restrictiveconditions on the parameters of the inverse problem. Observing that theselimitations can be due to the proximal algorithm in use, we study a relaxedversion of the PGD algorithm for minimizing the sum of a convex function and aweakly convex one. When plugged with a relaxed proximal denoiser, we show thatthe proposed PnP-$\alpha$PGD algorithm converges for a wider range ofregularization parameters, thus allowing more accurate image restoration.

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