SVRG meets AdaGrad: painless variance reduction

Abstract

Variance reduction (VR) methods for finite-sum minimization typically require the knowledge of problem-dependent constants that are often unknown and difficult to estimate. To address this, we use ideas from adaptive gradient methods to propose AdaSVRG, which is a more-robust variant of SVRG, a common VR method. AdaSVRG uses AdaGrad, a common adaptive gradient method, in the inner loop of SVRG, making it robust to the choice of step-size. When minimizing a sum of n smooth convex functions, we prove that a variant of AdaSVRG requires \(\tilde{O}(n + 1/\epsilon )\) gradient evaluations to achieve an \(O(\epsilon )\)-suboptimality, matching the typical rate, but without needing to know problem-dependent constants. Next, we show that the dynamics of AdaGrad exhibit a two-phase behavior – the step-size remains approximately constant in the first phase, and then decreases at a \(O\left( {1}/{\sqrt{t}}\right)\) rate. This result maybe of independent interest, and allows us to propose a heuristic that adaptively determines the length of each inner-loop in AdaSVRG. Via experiments on synthetic and real-world datasets, we validate the robustness and effectiveness of AdaSVRG, demonstrating its superior performance over standard and other “tune-free” VR methods.