HeteRSGD: Tackling Heterogeneous Sampling Costs via Optimal Reweighted Stochastic Gradient Descent

One implicit assumption in current stochastic gradient descent (SGD) algorithms is the identical cost for sampling each component function of the finite-sum objective. However, there are applications where the costs differ substantially, for which SGD schemes with uniform sampling invoke a high sampling load. We investigate the use of importance sampling (IS) as a cost saver in this setting, in contrast to its traditional use for variance reduction. The key ingredient is a novel efficiency metric for IS that advocates low sampling costs while penalizing high gradient variances. We then propose HeteRSGD, an SGD scheme that performs gradient sampling according to optimal probability weights stipulated by the metric, and establish theories on its optimal asymptotic and finite-time convergence rates among all possible IS-based SGD schemes. We show that the relative efficiency gain of HeteRSGD can be arbitrarily large regardless of the problem dimension and number of components. Our theoretical results are validated numerically for both convex and nonconvex problems.