Variance-Based Extragradient Methods with Line Search for Stochastic Variational Inequalities

Abstract

We propose dynamic sampled stochastic approximated (DS-SA) extragradient methods for stochastic variational inequalities (SVI) that are robust with respect to an unknown Lipschitz constant L. We propose, to the best of our knowledge, the first provably convergent robust SA method with variance reduction, either for SVIs or stochastic optimization, assuming just an unbiased stochastic oracle and a large sample regime. This widens the applicability and improves, up to constants, the desired efficient acceleration of previous variance reduction methods, all of which still assume knowledge of L (and, hence, are not robust against its estimate). Precisely, compared to the iteration and oracle complexities of O(ǫ −2 ) of previous robust methods with a small stepsize policy, our robust method uses a DS-SA line search scheme obtaining the faster iteration complexity of O(ǫ −1 ) with oracle complexity of (ln L)O(dǫ −2 ) (up to log factors on ǫ −1 ) for a d-dimensional space. This matches, up to constants, the sample complexity of the sample average approximation estimator which does not assume additional problem information (such as L). Differently from previous robust methods for ill-conditioned problems, we allow an unbounded feasible set and an oracle with multiplicative noise (MN) whose variance is not necessarily uniformly bounded. These properties are appreciated in our complexity estimates which depend only on L and local variances or forth moments at solutions x * . The robustness and variance reduction properties of our DS-SA line search scheme come at the expense of nonmartingale-like dependencies (NMD) due to the needed inner statistical estimation of a lower bound for L. In order to handle a NMD and a MN, our proofs rely on a novel localization argument based on empirical process theory. Additionally, we propose a second provable convergent method for SVIs over the wider class of Hölder continuous operators without any knowledge of its endogenous parameters.

On variance reduction for stochastic smooth convex optimization with multiplicative noise

“…Their complexity bounds typically grow with the oracle variance σ 2 in (8). See [8,15,58,20,25,26] and references therein. An essential point related to (Q) is if such increased effort in computation per iteration used is worth.…”

“…using (23) with a := y t+1 , b := z t+1 and c := x * . Now we sum the identities (24)- (26) and use the result in the right hand side of (22) obtaining…”

“…See e.g. [29,32,24,25,26,5] and references therein. VI is a framework which generalizes unconstrained system of equations and the first order necessary condition of constrained minimization (including a broader class of problems such as equilibrium, saddle-point and complementarity problems).…”

Open Problem—Iterative Schemes for Stochastic Optimization: Convergence Statements and Limit Theorems

“…When applying the line search, backtracking introduces a non-Martingale difference sequence, and a direct application of the law of large numbers is not possible. To overcome this challenge, empirical process theory is utilized to analyze the resulting processes (Iusem et al 2019), allowing for deriving optimal iteration complexity 2(1/ ) and near-optimal oracle complexity 2(1/ 2 ) (also see Paquette and Scheinberg (2018)).…”

A Variable Sample-Size Stochastic Quasi-Newton Method for Smooth and Nonsmooth Stochastic Convex Optimization

“…Consequently, standard analysis fails and one has to appeal to tools such as empirical process theory (cf. [15]). This remains the focus of future work.…”