Minimisation de sommes finies avec le gradient moyen stochastique

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‾‾√)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(𝜌𝑘)O(ρk) for 𝜌<1ρ<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.

Remerciements

Nous tenons à remercier les réviseurs anonymes pour leurs nombreux commentaires utiles. Ce travail a été partiellement soutenu par le Conseil européen de la recherche (SIERRA-ERC-239993) et une bourse de recherche Google. Mark Schmidt est également soutenu par une bourse postdoctorale du Conseil de recherches en sciences naturelles et en génie du Canada.