Martin Morin

Martin Morin, Pontus Giselsson

Variance-reduced stochastic gradient methods have gained popularity in recent times. Several variants exist with different strategies for the storing and sampling of gradients and this work concerns the interactions between these two aspects. We present a general proximal variance-reduced gradient method and analyze it under strong convexity assumptions. Special cases of the algorithm include SAGA, L-SVRG and their proximal variants. Our analysis sheds light on epoch-length selection and the need to balance the convergence of the iterates with how often gradients are stored. The analysis improves on other convergence rates found in the literature and produces a new and faster converging sampling strategy for SAGA. Problem instances for which the predicted rates are the same as the practical rates are presented together with problems based on real world data.

Martin Morin, Pontus Giselsson

With the purpose of examining biased updates in variance-reduced stochastic gradient methods, we introduce SVAG, a SAG/SAGA-like method with adjustable bias. SVAG is analyzed in a cocoercive root-finding setting, a setting which yields the same results as in the usual smooth convex optimization setting for the ordinary proximal-gradient method. We show that the same is not true for SVAG when biased updates are used. The step-size requirements for when the operators are gradients are significantly less restrictive compared to when they are not. This highlights the need to not rely solely on cocoercivity when analyzing variance-reduced methods meant for optimization. Our analysis either match or improve on previously known convergence conditions for SAG and SAGA. However, in the biased cases they still do not correspond well with practical experiences and we therefore examine the effect of bias numerically on a set of classification problems. The choice of bias seem to primarily affect the early stages of convergence and in most cases the differences vanish in the later stages of convergence. However, the effect of the bias choice is still significant in a couple of cases.