Abstract
The goal is to modify the known method of mirror descent (MD) in convex optimization, which having been proposed by Nemirovsky and Yudin in 1979 and generalized the standard gradient method. To start, the paper shows the idea of a new, so-called inertial MD method with the example of a deterministic optimization problem in continuous time. In particular, in the Euclidean case, the heavy ball method by Polyak is realized. It is noted that the new method does not use additional averaging of the points. Then, a discrete algorithm of inertial MD is described and the upper bound on error in objective function is proved. Finally, inertial MD randomized algorithm for finding a principal eigenvector of a given stochastic matrix (i.e., for solving a well known PageRank problem) is treated. Particular numerical example illustrates the general decrease of the error in time and corroborates theoretical results.
Partially supported by the Russian Science Foundation grant No 16–11–10015.
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Notes
- 1.
We are talking about the concept of an oracle of the first order in the optimization problem under consideration (either deterministic problem, when \(\xi _t\equiv 0\), or stochastic one, under \(\mathbb {E}\{\xi _t\}\equiv 0\)) [3].
- 2.
Below we mean \(\nabla _{x}^{}Q(x,Z_k)\) be the subgradient which are measurable functions defined on \({ X }\times \mathcal {Z}\) such that, for any \({ x }\in { X }\), the expectation \(\mathbb {E}u_k({ x })\) belongs to subdifferential \(\partial f(x)\).
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Acknowledgments
The work was partially supported by the Russian Science Foundation grant 16–11–10015. The author thanks B.T. Polyak for his attention to this work and A. Juditsky for important discussions and sending reference [9].
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Nazin, A. (2017). Algorithms of Inertial Mirror Descent in Stochastic Convex Optimization Problems. In: Rykov, V., Singpurwalla, N., Zubkov, A. (eds) Analytical and Computational Methods in Probability Theory. ACMPT 2017. Lecture Notes in Computer Science(), vol 10684. Springer, Cham. https://doi.org/10.1007/978-3-319-71504-9_31
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DOI: https://doi.org/10.1007/978-3-319-71504-9_31
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