Minimizing Finite Sums with the Stochastic Average Gradient
Abstract¶
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. The convergence rate is improved from O(1/k^{1/2}) to O(1/k) in general, and when the sum is strongly-convex the convergence rate is improved from the sub-linear O(1/k) to a linear convergence rate of the form O(p^k) for p \textless{} 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. 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.
Summary¶
Gives the extended analysis of SAG for smooth finite-sum convex objectives, proving faster rates than black-box stochastic gradient methods and linear convergence under strong convexity. Compared with the shorter NeurIPS paper, this version clarifies the Lyapunov-style proof, step-size behavior, storage tradeoffs, and non-uniform sampling ideas that made SAG a canonical variance-reduction method.