Multi-Iteration Stochastic Optimizers

Abstract

We here introduce Multi-Iteration Stochastic Optimizers, a novel class of first-order stochastic optimizers where the coefficient of variation of the mean gradient approximation, its relative statistical error, is estimated and controlled using successive control variates along the path of iterations. By exploiting the correlation between iterates, control variates may reduce the estimator's variance so that an accurate estimation of the mean gradient becomes computationally affordable. We name the estimator of the mean gradient Multi-Iteration stochastiC Estimator-MICE. In principle, MICE can be flexibly coupled with any first-order stochastic optimizer given its non-intrusive nature. Optimally, our generic algorithm decides whether to drop a particular iteration out of the control variates hierarchy, restart it, or clip it somewhere in the hierarchy, discarding previous iterations. We present a simplified study of the convergence and complexity of Multi-Iteration Stochastic Optimizers for strongly-convex and L-smooth functions. Motivated by this analysis, we provide a generic step-size choice for Stochastic Gradient Descent (SGD) with MICE, which, combined with the above characteristics, yields an efficient and robust stochastic optimizer. To assess the efficiency of MICE, we present several examples in which we use SGD-MICE and Adam-MICE. We include one example based on a stochastic adaptation of the Rosenbrock function and logistic regression training for various datasets. When compared to SGD, SAG, SAGA, SRVG, and SARAH methods, the Multi-Iteration Stochastic Optimizers reduced, without the need to tune parameters for each example, the gradient sampling cost in all cases tested, also lowering the total runtime in some cases.