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Why gradient clipping accelerates training: A theoretical justification for adaptivity (1905.11881v2)

Published 28 May 2019 in math.OC and cs.LG

Abstract: We provide a theoretical explanation for the effectiveness of gradient clipping in training deep neural networks. The key ingredient is a new smoothness condition derived from practical neural network training examples. We observe that gradient smoothness, a concept central to the analysis of first-order optimization algorithms that is often assumed to be a constant, demonstrates significant variability along the training trajectory of deep neural networks. Further, this smoothness positively correlates with the gradient norm, and contrary to standard assumptions in the literature, it can grow with the norm of the gradient. These empirical observations limit the applicability of existing theoretical analyses of algorithms that rely on a fixed bound on smoothness. These observations motivate us to introduce a novel relaxation of gradient smoothness that is weaker than the commonly used Lipschitz smoothness assumption. Under the new condition, we prove that two popular methods, namely, \emph{gradient clipping} and \emph{normalized gradient}, converge arbitrarily faster than gradient descent with fixed stepsize. We further explain why such adaptively scaled gradient methods can accelerate empirical convergence and verify our results empirically in popular neural network training settings.

Citations (410)

Summary

  • The paper introduces a novel smoothness condition that adapts to training dynamics, challenging traditional constant assumptions on gradient behavior.
  • It shows that gradient clipping and normalized gradient methods achieve faster convergence than standard gradient descent in both theory and empirical tests.
  • Empirical validations confirm that adaptive gradient methods enhance neural network training efficiency, informing the design of improved optimization algorithms.

Theoretical Justification for Gradient Clipping in Neural Network Training

The paper "Why gradient clipping accelerates training: A theoretical justification for adaptivity" offers an insightful theoretical exploration into the role of gradient clipping during the training of deep neural networks. Authored by a team from MIT, this work addresses a gap in existing theoretical frameworks by proposing a refined analysis of gradient smoothness and its impact on the convergence properties of training algorithms.

Key Contributions

The central contribution of this paper is the introduction of a novel smoothness condition that adapts to practical scenarios encountered in neural network training. Traditional analyses often assume that gradient smoothness is a constant, bounded by a Lipschitz condition. However, empirical observations by Zhang et al. reveal that gradient smoothness exhibits significant variability throughout the training process. Moreover, this variability correlates positively with the gradient norm, challenging the fixed bound assumption prevalent in existing literature.

In response to these findings, the authors propose a relaxation of the smoothness assumption. This relaxation enables a more faithful representation of empirical training dynamics. Utilizing this new condition, the paper provides proofs that both gradient clipping and normalized gradient methods achieve faster convergence compared to standard gradient descent with a fixed stepsize.

Empirical Verification and Results

The authors back their theoretical findings with a series of empirical validations within commonly used neural network training settings. The results demonstrate that adaptively scaled gradient methods, such as gradient clipping, can indeed enhance empirical convergence rates. These findings substantiate the theoretical claims and underline the practical utility of the proposed smoothness condition.

Implications and Future Directions

The implications of this research are twofold:

  1. Practical Impact: The theoretical justification for adaptive gradient methods, specifically gradient clipping, can guide the design of more efficient training regimes in deep learning architectures. This is particularly significant given the escalating complexity and resource demand in training modern neural networks.
  2. Theoretical Advancement: By relaxing the traditional Lipschitz smoothness assumption, this paper paves the way for more nuanced analyses of optimization algorithms in machine learning. Future research may extend this framework to other adaptive methods and explore its applicability across different model architectures and learning paradigms.

Overall, this paper contributes a robust theoretical foundation for understanding and leveraging gradient clipping in neural network training. As the field progresses, the insights from this work could inform both the development of new training algorithms and the refinement of existing ones.

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