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Recently, a number of learning-based optimization methods that combinedata-driven architectures with the classical optimization algorithms have beenproposed and explored, showing superior empirical performance in solvingvarious ill-posed inverse problems, but there is still a scarcity of rigorousanalysis about the convergence behaviors of learning-based optimization. Inparticular, most existing analyses are specific to unconstrained problems butcannot apply to the more general cases where some variables of interest aresubject to certain constraints. In this paper, we propose DifferentiableLinearized ADMM (D-LADMM) for solving the problems with linear constraints.Specifically, D-LADMM is a K-layer LADMM inspired deep neural network, which isobtained by firstly introducing some learnable weights in the classicalLinearized ADMM algorithm and then generalizing the proximal operator to somelearnable activation function. Notably, we rigorously prove that there exist aset of learnable parameters for D-LADMM to generate globally convergedsolutions, and we show that those desired parameters can be attained bytraining D-LADMM in a proper way. To the best of our knowledge, we are thefirst to provide the convergence analysis for the learning-based optimizationmethod on constrained problems.

Differentiable Linearized ADMM