Unified Optimal Analysis of the (Stochastic) Gradient Method

In this note we give a simple proof for the convergence of stochastic gradient (SGD) methods on $\mu$-convex functions under a (milder than standard) $L$-smoothness assumption. We show that for carefully chosen stepsizes SGD converges after $T$ iterations as $O\left( LR^2 \exp \bigl[-\frac{\mu}{4L}T...

Full description

Saved in:
Bibliographic Details
Main Author Stich, Sebastian U
Format Journal Article
LanguageEnglish
Published 09.07.2019
Subjects
Computer Science - Learning
Computer Science - Numerical Analysis
Mathematics - Numerical Analysis
Mathematics - Optimization and Control
Statistics - Machine Learning
Online AccessGet full text

Cover

More Information
Summary:In this note we give a simple proof for the convergence of stochastic gradient (SGD) methods on $\mu$-convex functions under a (milder than standard) $L$-smoothness assumption. We show that for carefully chosen stepsizes SGD converges after $T$ iterations as $O\left( LR^2 \exp \bigl[-\frac{\mu}{4L}T\bigr] + \frac{\sigma^2}{\mu T} \right)$ where $\sigma^2$ measures the variance in the stochastic noise. For deterministic gradient descent (GD) and SGD in the interpolation setting we have $\sigma^2 =0$ and we recover the exponential convergence rate. The bound matches with the best known iteration complexity of GD and SGD, up to constants.
DOI:10.48550/arxiv.1907.04232