Understanding Adam Requires Better Rotation Dependent Assumptions

Tianyue H. Zhang, Lucas Maes, Alan Milligan, Alexia Jolicoeur-Martineau, Ioannis Mitliagkas, Damien Scieur, Simon Lacoste-Julien, Charles Guille-Escuret

Keywords: Optimization, Adam, Deep Learning, Rotation Dependency

TL;DR: We observe that Adam’s performance in training transformers degrades differently under different types of random rotations of the objective function, highlighting the need for new, basis-dependent theory to fully understand Adam’s success.

Abstract: Despite its widespread adoption, Adam's advantage over Stochastic Gradient Descent (SGD) lacks a comprehensive theoretical explanation. This paper investigates Adam's sensitivity to rotations of the parameter space. We observe that Adam's performance in training transformers degrades under random rotations of the parameter space, indicating a crucial sensitivity to the choice of basis in practice. This reveals that conventional rotation-invariant assumptions are insufficient to capture Adam's advantages theoretically. To better understand the rotation-dependent properties that benefit Adam, we also identify structured rotations that preserve or even enhance its empirical performance. We then examine the rotation-dependent assumptions in the literature and find that they fall short in explaining Adam's behavior across various rotation types. In contrast, we verify the orthogonality of the update as a promising indicator of Adam’s basis sensitivity, suggesting it may be the key quantity for developing rotation-dependent theoretical frameworks that better explain its empirical success.

Primary Area: Deep learning (e.g., architectures, generative models, optimization for deep networks, foundation models, LLMs)

Submission Number: 19268