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Stochastic incremental mirror descent algorithms with Nesterov smoothing

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Abstract

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

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Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The work of the first named author was supported by the German Research Foundation (DFG), project WA922/9-1. The work of the second named author was partially supported by the Austrian Science Fund (FWF), project M-2045, by the Hi! PARIS Center, and by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH. For useful discussions regarding this paper the authors are thankful to Radu Ioan Boţ and Axel Böhm, to whom we are also grateful for providing us the program codes for their numerical experiments from [10], and to Gert Wanka. Valuable comments and suggestions from two anonymous reviewers are gratefully acknowledged, too.

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  1. Faculty of Mathematics, Chemnitz University of Technology, Chemnitz, 09126, Germany

    Sandy Bitterlich

  2. Unité de Mathématiques Appliquées, ENSTA Paris, Institut Polytechnique de Paris, 91120, Palaiseau, France

    Sorin-Mihai Grad

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Bitterlich, S., Grad, SM. Stochastic incremental mirror descent algorithms with Nesterov smoothing. Numer Algor 95, 351–382 (2024). https://doi.org/10.1007/s11075-023-01574-1

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