Stochastic incremental mirror descent algorithms with Nesterov smoothing

• Published in: Numerical algorithms Vol. 95; no. 1; pp. 351 - 382
• Main Authors: Bitterlich, Sandy, Grad, Sorin-Mihai
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.01.2024; Springer Verlag
• Subjects: Algebra; Algorithms; Computer Science; Mathematics; Numeric Computing; Numerical Analysis; Original Paper; Theory of Computation; Incremental algorithm; Nesterov smoothing; Mirror descent method; Stochastic algorithm; Proximal point algorithm; Location problem; PET image reconstructions; stochastic algorithm; incremental algorithm; proximal point algorithm
• ISSN: 1017-1398; 1572-9265; 1572-9265
• DOI: 10.1007/s11075-023-01574-1

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