Don't Jump Through Hoops and Remove Those Loops: SVRG and Katyusha are Better Without the Outer Loop

Abstract

The stochastic variance-reduced gradient method (SVRG) and its acceleratedvariant (Katyusha) have attracted enormous attention in the machine learningcommunity in the last few years due to their superior theoretical propertiesand empirical behaviour on training supervised machine learning models via theempirical risk minimization paradigm. A key structural element in both of thesemethods is the inclusion of an outer loop at the beginning of which a full passover the training data is made in order to compute the exact gradient, which isthen used to construct a variance-reduced estimator of the gradient. In thiswork we design {\em loopless variants} of both of these methods. In particular,we remove the outer loop and replace its function by a coin flip performed ineach iteration designed to trigger, with a small probability, the computationof the gradient. We prove that the new methods enjoy the same superiortheoretical convergence properties as the original methods. However, wedemonstrate through numerical experiments that our methods have substantiallysuperior practical behavior.