  Learning-to-optimize is an emerging framework that seeks to speed up the
solution of certain optimization problems by leveraging training data. Learned
optimization solvers have been shown to outperform classical optimization
algorithms in terms of convergence speed, especially for convex problems. Many
existing data-driven optimization methods are based on parameterizing the
update step and learning the optimal parameters (typically scalars) from the
available data. We propose a novel functional parameterization approach for
learned convex optimization solvers based on the classical mirror descent (MD)
algorithm. Specifically, we seek to learn the optimal Bregman distance in MD by
modeling the underlying convex function using an input-convex neural network
(ICNN). The parameters of the ICNN are learned by minimizing the target
objective function evaluated at the MD iterate after a predetermined number of
iterations. The inverse of the mirror map is modeled approximately using
another neural network, as the exact inverse is intractable to compute. We
derive convergence rate bounds for the proposed learned mirror descent (LMD)
approach with an approximate inverse mirror map and perform extensive numerical
evaluation on various convex problems such as image inpainting, denoising,
learning a two-class support vector machine (SVM) classifier and a multi-class
linear classifier on fixed features.
