Stochastic incremental mirror descent algorithms with Nesterov smoothing

Sandy Bitterlich (1) , Sorin-Mihai Grad (2, 3, 4)

1 Chemnitz University of Technology / Technische Universität Chemnitz
2 UMA - Unité de Mathématiques Appliquées
3 ENSTA Paris - École Nationale Supérieure de Techniques Avancées
4 OC - Optimisation et commande

For minimizing a sum of finitely many proper, convex and lower semicontinuous functions over a nonempty closed convex set in an Euclidean space we propose a stochastic incremental mirror descent algorithm constructed by means of the Nesterov smoothing. Further we modify the algorithm in order to minimize over a nonempty closed convex set in an Euclidean space a sum of finitely many proper, convex and lower semicontinuous functions composed with linear operators. Next a stochastic incremental mirror descent Bregman-proximal scheme with Nesterov smoothing is proposed in order to minimize over a nonempty closed convex set in an Euclidean space a sum of finitely many proper, convex and lower semicontinuous functions and a prox-friendly proper, convex and lower semicontinuous function. Different to the previous contributions from the literature on mirror descent methods for minimizing sums of functions, we do not require these to be (Lipschitz) continuous or differentiable. Applications in Logistics, Tomography and Machine Learning modelled as optimization problems illustrate the theoretical achievements.

Mots clés: Mirror descent method; stochastic algorithm; Nesterov smoothing; Incremental Algorithm; Proximal Algorithm; PET - Positron Emission Tomography; Support Vector Machines SVM

Domaines: Mathématiques [math]

Sandy Bitterlich, Sorin-Mihai Grad. Stochastic incremental mirror descent algorithms with Nesterov smoothing. 2021. ⟨hal-03428808v1⟩