Primal–Dual Mirror Descent Method for Constraint Stochastic Optimization Problems
- 37 Downloads
Extension of the mirror descent method developed for convex stochastic optimization problems to constrained convex stochastic optimization problems (subject to functional inequality constraints) is studied. A method that performs an ordinary mirror descent step if the constraints are insignificantly violated and performs a mirror descent step with respect to the violated constraint if this constraint is significantly violated is proposed. If the method parameters are chosen appropriately, a bound on the convergence rate (that is optimal for the given class of problems) is obtained and sharp bounds on the probability of large deviations are proved. For the deterministic case, the primal–dual property of the proposed method is proved. In other words, it is proved that, given the sequence of points (vectors) generated by the method, the solution of the dual method can be reconstructed up to the same accuracy with which the primal problem is solved. The efficiency of the method as applied for problems subject to a huge number of constraints is discussed. Note that the bound on the duality gap obtained in this paper does not include the unknown size of the solution to the dual problem.
Keywords:Mirror descent method convex stochastic optimization constrained optimization probability of large deviations randomization
We are grateful to Yu.E. Nesterov and A.S. Nemirovski for discussions of parts of this paper. We are also grateful to the reviewer for valuable remarks.
The work by A.V. Gasnikov was performed in the Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences and supported by the Russian Science Foundation (project no. 14-50-00150). The work by E.V. Gasnikova was supported by the Russian Foundation for Basic Research, project no. 15-31-20571-mol_a_ved.
- 1.A. S. Nemirovski and D. B. Yudin, Problem Complexity and Method Efficiency in Optimization, Interscience Series in Discrete Mathematics (Nauka, Moscow, 1979; Wiley, 1983), Vol. XV.Google Scholar
- 2.A. S. Anikin, A. V. Gasnikov, and A. Yu. Gornov, “Randomization and sparseness in huge-scale optimization problems using the mirror descent method as an example,” Trudy Mosk. Fiz.-Tekhn. Inst. 8 (1), 11–24 (2016). arXiv:1602.00594Google Scholar
- 3.K. Kim, Yu. Nesterov, V. Skokov, and B. Cherkasskii, “Efficient differentiation algorithms and extreme problems,” Ekon. Mat. Metody 20, 309–318 (1984).Google Scholar
- 5.A. V. Gasnikov, P. E. Dvurechensky, Yu. V. Dorn, and Yu. V. Maksimov, “Numerical methods for the problem of traffic flow equilibrium in the Backmann and the stable dynamics models,” Mat. Model. 28 (10), 40–64 (2016). arXiv:1506.00293Google Scholar
- 9.Yu. Nesterov, “New primal-dual subgradient methods for convex optimization problems with functional constraints,” Int. Workshop “Optimization and Statistical Learning”, Les Houches, France, 2015. http://lear.inrialpes.fr/workshop/osl2015/program.htmlGoogle Scholar
- 10.A. S. Anikin, A. V. Gasnikov, P. E. Dvurechensky, A. I. Tyurin, and A. V. Chernov, “Dual approaches to the minimization of strongly convex functionals with a simple structure under affine constraints,” Comput. Math. Math. Phys. 57, 1262–1275 (2017). arXiv:1602.01686Google Scholar
- 12.B. Cox, A. Juditsky, and A. Nemirovski, “Decomposition techniques for bilinear saddle point problems and variational inequalities with affine monotone operators on domains given by linear minimization oracles,”, 2015. arXiv:1506.02444Google Scholar
- 14.A. V. Gasnikov, E. A. Krymova, A. A. Lagunovskaya, I. N. Usmanova, and F. A. Fedorenko, “Stochastic online optimization. Single-point and multi-point non-linear multi-armed bandits. Convex and strongly-convex case,” Autom. Remote Control 78, 224–234 (2017). arXiv:1509.01679Google Scholar
- 15.J. C. Duchi, Introductory Lectures on Stochastic Optimization, IAS/Park City Mathematics Series.(2016), pp. 1–84. http://stanford.edu/~jduchi/PCMIConvex/Duchi16.pdfGoogle Scholar
- 16.Yu. Nesterov, “Subgradient methods for convex function with nonstandart growth properties,” 2016. http://www.mathnet.ru:8080/PresentFiles/16179/growthbm_nesterov.pdfGoogle Scholar
- 17.J. C. Duchi, S. Shalev-Shwartz, Y. Singer, and A. Tewari, “Composite objective mirror descent,” Proc. of COLT, 2010, pp. 14–26.Google Scholar
- 19.A. S. Anikin, A. V. Gasnikov, A. Yu. Gornov, D. I. Kamzolov, Yu. V. Maksimov, and Yu. E. Nesterov, “Efficient Numerical Solution of the PageRank problem for doubly sparse matrices,” Trudy Mosk. Fiz.-Tekhn. Inst., 7 (4), 74–94 (2015). arXiv:1508.07607Google Scholar
- 20.https://github.com/anastasiabayandina/MirrorGoogle Scholar
- 23.G. Lan and Z. Zhou, “Algorithms for stochastic optimization with expectation constraints,” 2016. http://pwp.gatech.edu/guanghui-lan/wp-content/uploads/sites/330/2016/08/SPCS8-19-16.pdfGoogle Scholar