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 \(\partial f(x)\).
References
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
Nesterov, Y.: Introductory Lectures on Convex Optimization. Kluwer, Boston (2004)
Nemirovskii, A.S., Yudin, D.B.: Problem Complexity and Method Efficiency in Optimization. Wiley, Chichester (1983)
Juditsky, A.B., Nazin, A.V., Tsybakov, A.B., Vayatis, N.: Recursive aggregation of estimators by the mirror descent algorithm with averaging. Probl. Inf. Transm. 41(4), 368–384 (2005)
Polyak, B.T.: Some methods of speeding up the convergence of iteration methods. USSR Comput. Math. Math. Phys. 4(5), 1–17 (1964)
Polyak, B.T.: Introduction Optimization. Optimization Software Inc., New York (1987)
Nazin, A.: Algorithms of inertial mirror descent in convex problems of stochastic optimization. ArXiv: 1705.01073 (2017)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Nesterov, Y., Shikhman, V.: Quasi-monotone subgradient methods for nonsmooth convex minimization. J. Optim. Theor. Appl. 165, 917–940 (2015)
Nazin, A.V.: Algorithms of inertial mirror descent in convex problems of stochastic optimization. Autom. Remote Control 79(1) (2018) accepted
Nazin, A.V., Polyak, B.T.: Randomized algorithm to determine the eigenvector of a stochastic matrix with application to the PageRank problem. Autom. Remote Control 72(2), 342–352 (2011)
Nemirovski, A., Juditsky, A., Lan, G., Shapiro, A.: Stochastic approximation approach to stochastic programming. http://www.optimization-online.org/DB_HTML/2007/09/1787.htm
Polyak, B.T., Timonina, A.V.: PageRank: new regularizations and simulation models. In: 18th IFAC World Congress, Milan, Italy, pp. 11202–11207 (2011)
Juditsky, A., Polyak, B.: Robust eigenvector of a stochastic matrix with application to pagerank. In: 51st IEEE Conference on Decision and Control, CDC 2012, Maui, Hawaii, USA, pp. 3171–3176 (2012)
Nazin, A.V.: Estimating the principal eigenvector of a stochastic matrix: mirror descent algorithms via game approach. In: The 49th IEEE Conference on Decision and Control (CDC 2010), Atlanta, Georgia USA, pp. 792–797 (2010)
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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