On the Convergence of Stochastic Gradient Descent with Adaptive Stepsizes

TL;DR

This paper provides a theoretical analysis of generalized AdaGrad stepsizes in stochastic gradient descent, demonstrating convergence guarantees and adaptive noise level handling in both convex and non-convex optimization.

Contribution

It offers the first convergence guarantees for generalized AdaGrad stepsizes in non-convex settings and shows their ability to adapt to gradient noise levels.

Findings

Achieves almost sure asymptotic convergence of gradients.

Automatically adapts to stochastic gradient noise levels.

Interpolates between O(1/T) and O(1/√T) convergence rates.

Abstract

Stochastic gradient descent is the method of choice for large scale optimization of machine learning objective functions. Yet, its performance is greatly variable and heavily depends on the choice of the stepsizes. This has motivated a large body of research on adaptive stepsizes. However, there is currently a gap in our theoretical understanding of these methods, especially in the non-convex setting. In this paper, we start closing this gap: we theoretically analyze in the convex and non-convex settings a generalized version of the AdaGrad stepsizes. We show sufficient conditions for these stepsizes to achieve almost sure asymptotic convergence of the gradients to zero, proving the first guarantee for generalized AdaGrad stepsizes in the non-convex setting. Moreover, we show that these stepsizes allow to automatically adapt to the level of noise of the stochastic gradients in both the convex and non-convex settings, interpolating between $O(1/T)$ and $O(1/\sqrt{T})$, up to logarithmic terms.