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Stochastic Heavy-Ball Method for Constrained Stochastic Optimization Problems

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Abstract

In this paper, we consider a heavy-ball method for the constrained stochastic optimization problem by focusing to the situation that the constraint set is specified as the intersection of possibly finitely many constraint sets. A variant algorithm of the stochastic heavy-ball method is proposed which will be incrementally processed by both the stochastic heavy-ball method and random constraint projection simultaneously. They converge almost surely to a solution of the suggested method is exhibited. Finally, a numerical experiment is discussed.

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References

  1. 1.

    Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)

  2. 2.

    Bauschke, H.H.: Projection algorithms: results and open problems. Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications. Stud. Comput Math, vol. 8, pp. 11–22. North-Holland (2001)

  3. 3.

    Cauchy, A. -L.: Méthode générale pour la résolution des systèmes d’équations simultanées. Comptes Rendus Hebd. Seances Acad. Sci. 25, 536–538 (1847)

  4. 4.

    Deutsch, F., Hundal, H.: The rate of convergence for the cyclic projections algorithm III: egularity of convex sets. J. Approx. Theory 155(2), 155–184 (2008)

  5. 5.

    Gubin, L.G., Polyak, B.T., Raik, E.V.: The method of projections for finding the common point of convex sets. USSR Comput. Math. Math. Phys. 7(6), 1–24 (1967)

  6. 6.

    Ghadimi, E., Feyzmahdavian, H.R., Johansson, M.: Global convergence of the heavy-ball method for convex optimization. arXiv:1412.7457.pdf(2014)

  7. 7.

    Hager, W.W., Zhang, H.: A survey of nonlinear conjugate gradient methods. Pac. J. Optim. 2(1), 35–58 (2006)

  8. 8.

    Johnson, R., Zhang, T.: Accelerating stochastic gradient descent using predictive variance reduction. Adv. NIPS 26, 315–323 (2013)

  9. 9.

    Loizou, N., Richtarik, P.: Linearly convergent stochastic heavy ball method for minimizing generalization error. In: 10th NIPS Workshop on Optimization for Machine Learning (NIPS 2017) (2017)

  10. 10.

    Nedic, A.: Random projection algorithms for convex set intersection problems. In: 49th IEEE Conference on Decision and Control (CDC), pp. 7655–7660 (2010)

  11. 11.

    Nemirovski, A., Juditsky, A., Lan, G., Shapiro, A.: Robust stochastic approximation approach to stochastic programming. SIAM J. Optim. 19(4), 1574–1609 (2008)

  12. 12.

    Neumann, J.V.: Functional Operators. Princeton University Press, Princeton (1950)

  13. 13.

    Nguyen, L.M., Nguyen, P.H., Dijk, M.V., Richtárik, P., Scheinberg, K., Takáč, M.: SGD and Hogwild! convergence without the bounded gradients assumption. Proceedings of the 35th International Conference on Machine Learning PMLR 80, 3747–3755 (2018)

  14. 14.

    Polyak, B.T.: Some methods of speeding up the convergence of iteration methods. USSR Comput. Math. Math. Phys. 4(5), 1–17 (1964)

  15. 15.

    Robbins, H., Monro, S.: A stochastic approximation method. Ann. Math. Statist. 22, 400–407 (1951)

  16. 16.

    Robbins, H., Siegmund, D.: A convergence theorem for non negative almost supermartingales and some applications. Optim. Methods Statistics, pp 233–257. Academic Press, N. Y. (1971)

  17. 17.

    Roux, N.L., Schmidt, M., Bach, F.: A stochastic gradient method with an exponential convergence rate for finite training sets. Adv. NIPS 4, 2663–2671 (2012)

  18. 18.

    Schmidt, M., Roux, N.L., Bach, F.: Minimizing finite sums with the stochastic average gradient. Technical report (2013)

  19. 19.

    Shanno, D.F., Phua, K.H.: Algorithm 500: Minimization of unconstrained multivariate functions [E4]. ACM Trans. Math. Softw. 2(1), 87–94 (1976)

  20. 20.

    Shanno, D.F.: On the convergence of a new conjugate gradient algorithm. SIAM J. Numer. Anal. 15(6), 1247–1257 (1978)

  21. 21.

    Strohmer, T., Vershynin, R.: A randomized Kaczmarz algorithm with exponential convergence. J. Fourier Anal. Appl. 15(2), 262–278 (2009)

  22. 22.

    Sun, T., Yin, P., Li, D., Huang, C., Guan, L., Jiang, H.: Non-ergodic convergence analysis of heavy-ball algorithms, arXiv:1811.01777(2018)

  23. 23.

    Wang, M., Bertsekas, D.P.: Stochastic first-order methods with random constraint projection. SIAM J. Optim. 26(1), 681–717 (2016)

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Correspondence to Narin Petrot.

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Promsinchai, P., Farajzadeh, A. & Petrot, N. Stochastic Heavy-Ball Method for Constrained Stochastic Optimization Problems. Acta Math Vietnam (2020). https://doi.org/10.1007/s40306-019-00357-y

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Keywords

  • Constrained stochastic optimization problem
  • Heavy-ball method
  • Random projection method
  • Converge almost surely

Mathematics Subject Classification (2010)

  • 90C15
  • 90C25
  • 90C06
  • 65K05