Can We Remove the Square-Root in Adaptive Gradient Methods? A Second-Order Perspective (2402.03496v10)

Abstract: Adaptive gradient optimizers like Adam(W) are the default training algorithms for many deep learning architectures, such as transformers. Their diagonal preconditioner is based on the gradient outer product which is incorporated into the parameter update via a square root. While these methods are often motivated as approximate second-order methods, the square root represents a fundamental difference. In this work, we investigate how the behavior of adaptive methods changes when we remove the root, i.e., strengthen their second-order motivation. Surprisingly, we find that such square-root-free adaptive methods close the generalization gap to SGD on convolutional architectures, while maintaining their root-based counterpart's performance on transformers. The second-order perspective also has practical benefits for developing non-diagonal methods that can incorporate arbitrary curvature approximations through the concept of preconditioner invariance. In contrast to root-based methods like Shampoo, root-free counterparts work well and fast with half-precision since they do not require numerically unstable matrix root decompositions and inversions. Overall, our findings provide new insights into the development of adaptive methods and raise important questions regarding the overlooked role of adaptivity in their success. (experiment code: https://github.com/yorkerlin/remove-the-square-root optimizer code: https://github.com/f-dangel/sirfshampoo)

• The paper presents a novel adaptive gradient method that removes the square-root operation, matching or exceeding performance of traditional optimizers.
• The authors provide a second-order theoretical framework that reinterprets the gradient outer product as an empirical Fisher matrix to support their approach.
• Empirical results show improved generalization on CNNs and efficient training on Transformers by reducing computational overhead.

Analysis of Adaptive Gradient Methods Without the Square Root

The paper "Can We Remove the Square-Root in Adaptive Gradient Methods?" presents a critical examination of adaptive gradient optimizers, particularly Adam and similar methods, and proposes an adaptation strategy that omits the computational square root in their updates. This investigation is deeply rooted in the context of modern training strategies used for deep learning models, especially Transformers and Convolutional Neural Networks (CNNs). The paper intricately explores both the theoretical aspects of these optimizers and the practical benefits of excluding the square root, offering a nuanced perspective on optimization in deep learning.

Summary of Main Contributions

The authors' primary contributions are twofold: empirical findings and theoretical advancements.

1. Empirical Observations: The paper provides extensive empirical evidence that square-root-free methods can match or even exceed their root-based counterparts. Particularly noteworthy is that while these adapted methods performed on par with traditional methods for training Transformers, they also narrowed the generalization gap typically observed when using stochastic gradient descent (SGD) on CNN architectures. By removing the square root, the methods maintain performance on architectures like vision transformers and generalize better on convolutional architectures.
2. Theoretical Framework: On the theoretical front, the authors present a second-order perspective that supports the removal of the square root. They reinterpret the gradient outer product as a variant of the empirical Fisher information matrix, which aligns with the expectations from second-order methods. This redefined approach not only provides an insightful connection between the Fisher approximation and Hessian matrices but also allows for robust operation in low-precision settings.
3. Computational Efficiency: In contrast to methods such as Shampoo, which rely on matrix square roots that are computationally intense and susceptible to numerical instabilities, the proposed methods thrive in low-precision environments. The elimination of square roots circumvents the complex matrix decompositions traditionally necessary, thereby reducing memory consumption and enhancing computational efficiency.

Implications and Future Directions

The paper opens up new research avenues in understanding the intrinsic role of adaptivity in the success of these optimization methods. The proposed methods demonstrate that adaptivity, rather than sign descent, could be pivotal in achieving superior performance in diverse architectures. This realization encourages a reevaluation of the foundational assumptions about how these methods function and thrive.

From a theoretical standpoint, further investigation into the disentangled roles of adaptivity and sign descent could offer deeper insights, possibly leading to even more efficient algorithms. Practically, the application of these findings can revolutionize training strategies for large models, where computational resources are a bottleneck.

Future research could also delve into incorporating these insights into distributed and parallel computing frameworks, as well as exploring their implications on emerging hardware accelerators. The potential for developing new variants of adaptive methods that exploit this understanding could signify substantial progress in the field.

This work thus serves as a thoughtful and technically sophisticated critique of the status quo, suggesting a refined approach that questions long-held assumptions in gradient-based optimization. As deep learning models continue to grow in complexity and size, these methods that balance performance and computational costs will be pivotal.