HomeAdam: Adam and AdamW Algorithms Sometimes Go Home to Obtain Better Provable Generalization

TL;DR

This paper analyzes the generalization properties of Adam and AdamW optimizers, proposing a new variant called HomeAdam(W) that achieves better theoretical generalization bounds and convergence rates, supported by numerical experiments.

Contribution

The paper provides the first theoretical analysis of Adam(W)'s generalization error and introduces HomeAdam(W), a novel optimizer with improved generalization and convergence guarantees.

Findings

HomeAdam(W) achieves $O(1/N)$ generalization error, better than Adam(W)-srf and existing Adam variants.

HomeAdam(W) has a faster convergence rate of $O(1/T^{1/4})$ compared to Adam(W)-srf.

Numerical experiments confirm the efficiency of HomeAdam(W) in practice.

Abstract

Adam and AdamW are a class of default optimizers for training deep learning models in machine learning. These adaptive algorithms converge faster but generalize worse compared to SGD. In fact, their proved generalization error $O(\frac{1}{\sqrt{N}})$ also is larger than $O(\frac{1}{N})$ of SGD, where $N$ denotes training sample size. Recently, although some variants of Adam have been proposed to improve its generalization, their improved generalizations are still unexplored in theory. To fill this gap, in the paper, we restudy generalization of Adam and AdamW via algorithmic stability, and first prove that Adam and AdamW without square-root (i.e., Adam(W)-srf) have a generalization error $O(\frac{\hat{\rho}^{-2T}}{N})$, where $T$ denotes iteration number and $\hat{\rho}>0$ denotes the smallest element of second-order momentum plus a small positive number. To improve generalization, we propose a class of efficient clever Adam (i.e., HomeAdam(W)) algorithms via sometimes returning momentum-based SGD. Moreover, we prove that our HomeAdam(W) have a smaller generalization error $O(\frac{1}{N})$ than $O(\frac{\hat{\rho}^{-2T}}{N})$ of Adam(W)-srf, since $\hat{\rho}$ is generally very small. In particular, it is also smaller than the existing $O(\frac{1}{\sqrt{N}})$ of Adam(W). Meanwhile, we prove our HomeAdam(W) have a faster convergence rate of $O(\frac{1}{T^{1/4}})$ than $O(\frac{\breve{\rho}^{-1}}{T^{1/4}})$ of the Adam(W)-srf, where $\breve{\rho}\leq\hat{\rho}$ also is very small. Extensive numerical experiments demonstrate efficiency of our HomeAdam(W) algorithms.