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Accelerated schemes for a class of variational inequalities

Abstract

We propose a novel stochastic method, namely the stochastic accelerated mirror-prox (SAMP) method, for solving a class of monotone stochastic variational inequalities (SVI). The main idea of the proposed algorithm is to incorporate a multi-step acceleration scheme into the stochastic mirror-prox method. The developed SAMP method computes weak solutions with the optimal iteration complexity for SVIs. In particular, if the operator in SVI consists of the stochastic gradient of a smooth function, the iteration complexity of the SAMP method can be accelerated in terms of their dependence on the Lipschitz constant of the smooth function. For SVIs with bounded feasible sets, the bound of the iteration complexity of the SAMP method depends on the diameter of the feasible set. For unbounded SVIs, we adopt the modified gap function introduced by Monteiro and Svaiter for solving monotone inclusion, and show that the iteration complexity of the SAMP method depends on the distance from the initial point to the set of strong solutions. It is worth noting that our study also significantly improves a few existing complexity results for solving deterministic variational inequality problems. We demonstrate the advantages of the SAMP method over some existing algorithms through our preliminary numerical experiments.

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Notes

  1. 1.

    When the maximum absolute values of P and Q are different, it is recommended to introduce weights \(\omega _x\) and \(\omega _y\) and set \(\Vert u\Vert :=\sqrt{\omega _x\Vert x\Vert _1^2 + \omega _y\Vert y\Vert _1^2}\) and \(\Vert \eta \Vert _*:=\sqrt{\Vert \eta _x\Vert _1^2/\omega _x + \Vert \eta _y\Vert _1^2/\omega _y}\). See “mixed setups” in Section 5 of [32] for the detailed derivations for best values of weights \(\omega _x\) and \(\omega _y\).

  2. 2.

    See the proof of Theorem 2 for the definition of the perturbation term in the SAMP algorithm, and Theorem 5.2 in [28] for the definition of the perturbation term in the MP algorithm.

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Correspondence to Guanghui Lan.

Additional information

Yunmei Chen is partially supported by NSF Grants DMS-1115568, IIP-1237814 and DMS-1319050. Guanghui Lan is partially supported by NSF Grants CMMI-1637473, CMMI-1637474, DMS-1319050 and ONR Grant N00014-16-1-2802. Part of the research was done while Yuyuan Ouyang was a Ph.D. student at the Department of Mathematics, University of Florida, and Yuyuan Ouyang is partially supported by AFRL Mathematical Modeling Optimization Institute.

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Chen, Y., Lan, G. & Ouyang, Y. Accelerated schemes for a class of variational inequalities. Math. Program. 165, 113–149 (2017). https://doi.org/10.1007/s10107-017-1161-4

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Keywords

  • Stochastic variational inequalities
  • Stochastic programming
  • Mirror-prox method
  • Extragradient method

Mathematics Subject Classification

  • 90C25
  • 90C15
  • 62L20
  • 68Q25