Improved variance reduction extragradient method with line search for stochastic variational inequalities

In this paper, we investigate the numerical methods for solving stochastic variational inequalities. Using line search scheme, we propose an improved variance based stochastic extragradient method with different step sizes in the prediction and correction steps. The range of correction step size which can guarantee the convergence is also given. For the initial line search step size of each iteration, an adaptive method is adopted. Rather than the same scale for each reduction, a proportional reduction related to the problem is used to meet the line search criteria. Under the assumptions of Lipschitz continuous, pseudo-monotone operator and independent identically distributed sampling, the iterative complexity and the oracle complexity are obtained. When estimating the upper bound of the second order moment of the martingale difference sequence, we give a more convenient and comprehensible proof instead of using the Burkholder-Davis-Gundy inequality. The proposed algorithm is applied to fractional programming problems and the l_2 regularized logistic regression problem. The numerical results demonstrate its superiority.