Differentiating Nonsmooth Solutions to Parametric Monotone Inclusion Problems

TL;DR

This paper establishes conditions under which solutions to parametric monotone inclusion problems are path differentiable, enabling automatic differentiation and broad applicability to convex and min-max problems.

Contribution

It introduces sufficient conditions for path differentiability of solutions to monotone inclusions, leveraging recent nonsmooth calculus and applicable to various fundamental problem settings.

Findings

Solutions are path differentiable under semialgebraic and strong monotonicity assumptions.

Solutions are differentiable almost everywhere, facilitating gradient-based methods.

The approach is compatible with automatic differentiation in practical applications.

Abstract

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