Understanding Adam Requires Better Rotation Dependent Assumptions

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 demonstrate that Adam's performance in training transformers degrades under random rotations of the parameter space, indicating a crucial sensitivity to the choice of basis. 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, evaluating their adequacy in explaining Adam's behavior across various rotation types. This work highlights the need for new, rotation-dependent theoretical frameworks to fully understand Adam's empirical success in modern machine learning tasks.

• The paper demonstrates how Adam’s efficiency can drop by up to 96% when the parameter space undergoes random rotations.
• The paper reveals that structured rotations, particularly those from SVD, can preserve or even enhance Adam's convergence.
• The paper critiques existing rotation-invariant assumptions and calls for rotation-aware frameworks to better explain optimizer behavior.

Understanding the Rotation Sensitivity of Adam

In this paper, the authors rigorously examine the Adaptive Moment Estimation (Adam) optimizer, which remains a staple in training large-scale neural networks, particularly transformers, despite the theoretical convergence issues identified in its early iterations. They focus on a previously underexplored facet: Adam's sensitivity to the parameter space's coordinate system. This study reveals that Adam's performance varies significantly when the objective function undergoes random rotations, suggesting that assumptions regarding optimizer performance must be reconsidered and potentially revised from a rotation-dependent perspective.

Key Insights and Findings

1. Rotation Sensitivity: The authors provide compelling empirical evidence showing that Adam's efficiency diminishes when the parameter space is subject to random global rotations. They illustrate this using LLMs such as GPT-2, where rotations introduced a performance slowdown of up to 96%. This highlights a stark contrast to SGD, which is rotation-equivariant, maintaining its optimization trajectory regardless of parameter space orientation.
2. Structured Rotations: The study identifies certain structured rotations, particularly those derived from Singular Value Decomposition (SVD) of gradient matrices, that can preserve or even enhance Adam's performance. This rotation-aware approach could be instrumental in improving Adam's convergence in practice.
3. Theoretical Framework Reevaluation: The authors assess the applicability of several existing theoretical assumptions used in analyzing optimization techniques. Their findings indicate that conventional rotation-invariant assumptions fail to adequately model Adam's behavior under rotation. They specifically critique the utility of L∞-bounded gradients, block-diagonal Hessian assumptions, and L∞-smoothness, highlighting gaps and potential areas for theoretical advancement and refinement.

Implications and Future Directions

The implications of this research are multifaceted, adding a new dimension to how optimization techniques are theoretically analyzed and practically applied:

• Practical Algorithms:

The insights on rotation sensitivity and structured rotations could inform the development of next-generation optimizers that are adaptable across different coordinate systems, potentially leading to enhanced performance in various machine learning applications.

• Theoretical Refinement:

The identification of inadequacies in current theoretical models opens a pathway for future studies to develop more nuanced, rotation-aware frameworks. This can enhance our understanding of optimization algorithms beyond their current scope, potentially improving convergence proofs and practical performance guarantees.

• Broader Application Domains:

While the focus is on Adam and its efficacy in training transformers, the results suggest examining other optimizers through a similar lens, possibly unearthing general principles of rotation dependence in optimization.

The paper underscores a critical need for further investigation into the foundational aspects of optimization in neural networks, urging a rethink of how these tools are understood in both theoretical and empirical contexts. As the AI community continues to grapple with increasingly complex models, such insights will be invaluable in driving forward both technical and theoretical advancements.