Differentiating Nonsmooth Solutions to Parametric Monotone Inclusion Problems

Abstract

We leverage path differentiability and a recent result on nonsmooth implicit\ndifferentiation calculus to give sufficient conditions ensuring that the\nsolution to a monotone inclusion problem will be path differentiable, with\nformulas for computing its generalized gradient. A direct consequence of our\nresult is that these solutions happen to be differentiable almost everywhere.\nOur approach is fully compatible with automatic differentiation and comes with\nassumptions which are easy to check, roughly speaking: semialgebraicity and\nstrong monotonicity. We illustrate the scope of our results by considering\nthree fundamental composite problem settings: strongly convex problems, dual\nsolutions to convex minimization problems and primal-dual solutions to min-max\nproblems.\n

Topics & Concepts