Minimizing finite sums with the stochastic average gradient

• Published in: Mathematical programming Vol. 162; no. 1-2; pp. 83 - 112
• Main Authors: Schmidt, Mark, Le Roux, Nicolas, Bach, Francis
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2017; Springer Nature B.V; Springer Verlag
• Subjects: Algorithms; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Computation; Computer Science; Convergence; Convex analysis; Full Length Paper; Machine Learning; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical models; Mathematics; Mathematics and Statistics; Mathematics of Computing; Methods; Numerical Analysis; Optimization; Optimization and Control; Regression analysis; Sag; Sampling; Statistics; Stochasticity; Strategy; Studies; Texts; Theoretical; 68Q25; 90C30; 65K05; Convex optimization; 90C25; 90C06; Stochastic gradient methods; 90C15; Convergence Rates; First-order methods; 62L20; Stochastic Gradient; Convex Optimization
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-016-1030-6

We analyze 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 ) 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 ( ρ k ) for ρ < 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. This extends our earlier work Le Roux et al. (Adv Neural Inf Process Syst, 2012 ), which only lead to a faster rate for well-conditioned strongly-convex problems. 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.