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.

翻译：本文提出了即插即用（PnP）算法收敛性的新证明。PnP方法是一种高效的迭代算法，用于解决图像逆问题，其通过将预训练去噪器插入近端算法（如近端梯度下降（PGD）或Douglas-Rachford分裂（DRS））来实现正则化。近期研究通过引入可精确表示为近端算子的去噪器来探索收敛性，但相应的PnP算法必须以步长等于1运行。此时，近端算法非凸收敛的步长条件转化为逆问题正则化参数的严格限制条件，这会严重降低算法的恢复能力。本文针对该局限提出两种解决方案：首先，我们给出PnP-DRS算法的新型收敛证明，该证明不对正则化参数施加任何限制；其次，我们考察一种松弛化PGD算法，使其能在更广泛的正则化参数范围内收敛。基于去模糊和超分辨率实验的数值研究表明，这两种方案均能提升图像恢复的精度。