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Algorithms of Inertial Mirror Descent in Convex Problems of Stochastic Optimization

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Abstract

A minimization problem for mathematical expectation of a convex loss function over given convex compact X ∈ RN is treated. It is assumed that the oracle sequentially returns stochastic subgradients for loss function at current points with uniformly bounded second moment. The aim consists in modification of well-known mirror descent method proposed by A.S. Nemirovsky and D.B. Yudin in 1979 and having extended the standard gradient method. In the beginning, the idea of a new so-called method of Inertial Mirror Descent (IMD) on example of a deterministic optimization problem in RN with continuous time is demonstrated. Particularly, in Euclidean case the method of heavy ball is realized; it is noted that the new method no use additional point averaging. Further on, a discrete IMD algorithm is described; the upper bound on error over objective function (i.e., of the difference between current mean losses and their minimum) is proved.

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Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    A. V. Nazin

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  1. A. V. Nazin
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Correspondence to A. V. Nazin.

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Original Russian Text © A.V. Nazin, 2018, published in Avtomatika i Telemekhanika, 2018, No. 1, pp. 100-112.

This paper was recommended for publication by A.I. Kibzun, a member of the Editorial Board

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Nazin, A.V. Algorithms of Inertial Mirror Descent in Convex Problems of Stochastic Optimization. Autom Remote Control 79, 78–88 (2018). https://doi.org/10.1134/S0005117918010071

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  • DOI: https://doi.org/10.1134/S0005117918010071

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