Towards Theoretically Understanding Why SGD Generalizes Better Than ADAM in Deep Learning

TL;DR

This paper provides a theoretical analysis explaining why SGD generalizes better than ADAM by examining their local convergence behaviors and the impact of gradient noise tails on escaping sharp minima.

Contribution

It introduces a Levy-driven SDE framework to analyze the escaping times from local basins, revealing why SGD better escapes sharp minima compared to ADAM.

Findings

SGD has smaller escaping times from sharp minima than ADAM.

Heavy-tailed gradient noise influences the stability and escape behavior of algorithms.

Experimental results support the heavy-tailed noise assumption and theoretical insights.

Abstract

It is not clear yet why ADAM-alike adaptive gradient algorithms suffer from worse generalization performance than SGD despite their faster training speed. This work aims to provide understandings on this generalization gap by analyzing their local convergence behaviors. Specifically, we observe the heavy tails of gradient noise in these algorithms. This motivates us to analyze these algorithms through their Levy-driven stochastic differential equations (SDEs) because of the similar convergence behaviors of an algorithm and its SDE. Then we establish the escaping time of these SDEs from a local basin. The result shows that (1) the escaping time of both SGD and ADAM~depends on the Radon measure of the basin positively and the heaviness of gradient noise negatively; (2) for the same basin, SGD enjoys smaller escaping time than ADAM, mainly because (a) the geometry adaptation in ADAM~via adaptively scaling each gradient coordinate well diminishes the anisotropic structure in gradient noise and results in larger Radon measure of a basin; (b) the exponential gradient average in ADAM~smooths its gradient and leads to lighter gradient noise tails than SGD. So SGD is more locally unstable than ADAM~at sharp minima defined as the minima whose local basins have small Radon measure, and can better escape from them to flatter ones with larger Radon measure. As flat minima here which often refer to the minima at flat or asymmetric basins/valleys often generalize better than sharp ones , our result explains the better generalization performance of SGD over ADAM. Finally, experimental results confirm our heavy-tailed gradient noise assumption and theoretical affirmation.