Differentiating Nonsmooth Solutions to Parametric Monotone Inclusion Problems

Jérôme Bolte (1) , Edouard Pauwels (2) , Antonio José Silveti-Falls (3, 4)

1 TSE-R - Toulouse School of Economics
2 IRIT-ADRIA - Argumentation, Décision, Raisonnement, Incertitude et Apprentissage
3 CVN - Centre de vision numérique
4 OPIS - OPtimisation Imagerie et Santé

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.

Keywords: Maximal monotone operator; monotone inclusion; generalized equation; implicit differentiation; dif- ferentiating solutions; Clarke subdifferential; generalized gradient; conservative field

Domains: Optimization and Control [math.OC]; Machine Learning [cs.LG]

Jérôme Bolte, Edouard Pauwels, Antonio José Silveti-Falls. Differentiating Nonsmooth Solutions to Parametric Monotone Inclusion Problems. SIAM Journal on Optimization, 2024, 34 (1), 27 p. ⟨10.1137/22M1541630⟩. ⟨hal-03900339⟩