Continuous-Time Analysis of Adaptive Optimization and Normalization

TL;DR

This paper introduces a continuous-time framework for Adam and AdamW optimizers, providing theoretical insights into their training dynamics, hyperparameter stability, and the implicit effects of normalization layers, leading to a new family of adaptive optimizers.

Contribution

It develops a continuous-time analysis of Adam, derives stable hyperparameter regions, and introduces k-Adam, a generalized optimizer incorporating multiple normalization steps.

Findings

Stable hyperparameter regions for Adam's convergence.

Implicit meta-adaptive effects of normalization layers.

Introduction of the k-Adam optimizer family.

Abstract

Adaptive optimization algorithms, particularly Adam and its variant AdamW, are fundamental components of modern deep learning. However, their training dynamics lack comprehensive theoretical understanding, with limited insight into why common practices -- such as specific hyperparameter choices and normalization layers -- contribute to successful generalization. This work presents a continuous-time formulation of Adam and AdamW, facilitating a tractable analysis of training dynamics that can shed light on such practical questions. We theoretically derive a stable region for Adam's hyperparameters $(\beta, \gamma)$ that ensures bounded updates, empirically verifying these predictions by observing unstable exponential parameter growth outside of this stable region. Furthermore, we theoretically justify the success of normalization layers by uncovering an implicit meta-adaptive effect of scale-invariant architectural components. This insight leads to an explicit optimizer, $2$-Adam, which we generalize to $k$-Adam -- an optimizer that applies an adaptive normalization procedure $k$ times, encompassing Adam (corresponding to $k=1$) and Adam with a normalization layer (corresponding to $k=2$). Overall, our continuous-time formulation of Adam facilitates a principled analysis, offering deeper understanding of optimal hyperparameter choices and architectural decisions in modern deep learning.