Differentiable Linearized ADMM

Recently, a great many learning-based optimization methods that combine data-driven architectures with the classical optimization algorithms have been proposed and explored, showing superior empirical performance in solving various ill-posed inverse problems. However, there is still a scarcity of rigorous analysis about the convergence behaviors of learning-based optimization. In particular, most existing theories are specific to unconstrained problems but cannot apply to the more general cases where some variables of interest are subject to certain constraints. In this paper, we propose Differentiable Linearized ADMM (D-LADMM) for solving the problems with linear constraints. Specifically, D-LADMM is a K-layer LADMM inspired deep neural network, which is obtained by firstly introducing some learnable weights in the classical Linearized ADMM algorithm and then generalizing the proximal operator to some learnable activation function. Notably, we mathematically prove that there exist a set of learnable parameters for D-LADMM to generate globally converged solutions, and we show that those desired parameters can be attained by training D-LADMM in a proper way. To the best of our knowledge, we are the first one to provide the convergence analysis for the learning-based optimization method on constrained problems. Experiments on simulative and real applications verify the superiorities of D-LADMM over LADMM.