A Primal-Dual Algorithm for Image Reconstruction with Input-Convex Neural Network Regularizers

We address the optimization problem in a data-driven variational reconstruction framework, where the regularizer is parametrized by an input-convex neural network. While gradient-based methods are commonly used to solve such problems, they struggle to effectively handle nonsmooth prob lems, which often leads to slow convergence. Moreover, the nested structure of the neural network complicates the application of standard nonsmooth optimization techniques, such as proximal al gorithms. To overcome these challenges, we reformulate the problem and eliminate the network's nested structure. By relating this reformulation to epigraphical projections of the activation func tions, we transform the problem into a convex optimization problem that can be efficiently solved using a primal-dual algorithm. We also prove that this reformulation is equivalent to the original variational problem. Through experiments on several imaging tasks, we show that the proposed approach not only outperforms subgradient methods and even accelerated methods in the smooth setting but also facilitates the training of the regularizer itself.