Almost sure convergence rates for Stochastic Gradient Descent and Stochastic Heavy Ball

Othmane Sebbouh (1, 2, 3, 4) , Robert M Gower (5) , Aaron Defazio (6)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 CREST - Centre de Recherche en Économie et Statistique
3 ENSAE - Ecole Nationale de la Statistique et de l'Analyse Economique
4 CNRS - Centre National de la Recherche Scientifique
5 Télécom ParisTech
6 FAIR - Facebook AI Research [New York]

We study stochastic gradient descent (SGD) and the stochastic heavy ball method (SHB, otherwise known as the momentum method) for the general stochastic approximation problem. For SGD, in the convex and smooth setting, we provide the first almost sure asymptotic convergence rates for a weighted average of the iterates. More precisely, we show that the convergence rate of the function values is arbitrarily close to o(1/ √ k), and is exactly o(1/k) in the so-called overparametrized case. We show that these results still hold when using stochastic line search and stochastic Polyak stepsizes, thereby giving the first proof of convergence of these methods in the non-overparametrized regime. Using a substantially different analysis, we show that these rates hold for SHB as well, but at the last iterate. This distinction is important because it is the last iterate of SGD and SHB which is used in practice. We also show that the last iterate of SHB converges to a minimizer almost surely. Additionally, we prove that the function values of the deterministic HB converge at a o(1/k) rate, which is faster than the previously known O(1/k). Finally, in the nonconvex setting, we prove similar rates on the lowest gradient norm along the trajectory of SGD.

Domaines: Informatique [cs]; Mathématiques [math]

Othmane Sebbouh, Robert M Gower, Aaron Defazio. Almost sure convergence rates for Stochastic Gradient Descent and Stochastic Heavy Ball. 2021. ⟨hal-03135145⟩