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Stochastic first order methods in smooth convex optimization

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  • DEVOLDER, Olivier

    () (Université catholique de Louvain, CORE, B-1348 Louvain-la-Neuve, Belgium)

Abstract

In this paper, we are interested in the development of efficient first-order methods for convex optimization problems in the simultaneous presence of smoothness of the objective function and stochasticity in the first-order information. First, we consider the Stochastic Primal Gradient method, which is nothing else but the Mirror Descent SA method applied to a smooth function and we develop new practical and efficient stepsizes policies. Based on the machinery of estimates sequences functions, we develop also two new methods, a Stochastic Dual Gradient Method and an accelerated Stochastic Fast Gradient Method. Convergence rates on average, probabilities of large deviations and accuracy certificates are studied. All of these methods are designed in order to decrease the effect of the stochastic noise at an unimprovable rate and to be easily implementable in practice (the practical efficiency of our method is confirmed by numerical experiments). Furthermore, the biased case, when the oracle is not only stochastic but also affected by a bias is considered for the first time in the literature.

Suggested Citation

  • DEVOLDER, Olivier, 2011. "Stochastic first order methods in smooth convex optimization," CORE Discussion Papers 2011070, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  • Handle: RePEc:cor:louvco:2011070
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    File URL: http://uclouvain.be/cps/ucl/doc/core/documents/coredp2011_70web.pdf
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    References listed on IDEAS

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    1. Fleurbaey,Marc & Maniquet,François, 2011. "A Theory of Fairness and Social Welfare," Cambridge Books, Cambridge University Press, number 9780521715348, May.
    2. de la Croix, David & Gosseries, Axel, 2012. "The natalist bias of pollution control," Journal of Environmental Economics and Management, Elsevier, vol. 63(2), pages 271-287.
    3. Duranton, Gilles & Martin, Philippe & Mayer, Thierry & Mayneris, Florian, 2010. "The Economics of Clusters: Lessons from the French Experience," OUP Catalogue, Oxford University Press, number 9780199592203.
    4. DEVOLDER, Olivier & GLINEUR, François & NESTEROV, Yurii, 2011. "First-order methods of smooth convex optimization with inexact oracle," CORE Discussion Papers 2011002, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Belleflamme,Paul & Peitz,Martin, 2010. "Industrial Organization," Cambridge Books, Cambridge University Press, number 9780521681599, November.
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