SCAIM Seminar: Mark Schmidt

We propose the stochastic average gradient (SAG) method for optimizing the sum of a finite number of smooth convex functions. Like stochastic gradient (SG) methods, the SAG method's iteration cost is independent of the number of terms in the sum. However, by incorporating a memory of previous gradient values the SAG method achieves a faster convergence rate than black-box SG methods. Specifically, under standard assumptions the convergence rate is improved from O(1/k) to a linear convergence rate of the form O(p^k) for some p 1. Further, in many cases the convergence rate of the new method is also faster than black-box deterministic gradient methods, in terms of the number of gradient evaluations. Beyond these theoretical results, the algorithm also has a variety of appealing practical properties: it supports regularization and sparse datasets, it allows an adaptive step-size and has a termination criterion, it allows mini-batches, and its performance can be further improved by non-uniform sampling. Numerical experiments indicate that the new algorithm often dramatically outperforms existing SG and deterministic gradient methods, and that the performance may be further improved through the use of non-uniform sampling strategies.