Analysis of Gradient Clipping and Adaptive Scaling with a Relaxed Smoothness Condition
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We provide a theoretical explanation for the fast convergence of gradient clipping and adaptively scaled gradient methods commonly used in neural network training. Our analysis is based on a novel relaxation of gradient smoothness conditions that is weaker than the commonly used Lipschitz smoothness assumption. We validate the new smoothness condition in experiments on large-scale neural network training applications where adaptively-scaled methods have been empirically shown to outperform standard gradient based algorithms. Under this new smoothness condition, we prove that two popular adaptively scaled methods, gradient clipping and normalized gradient, converge faster than the theoretical lower bound of fixed-step gradient descent. We verify this fast convergence empirically in neural network training for language modeling and image classification.
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