Step Size Matters in Deep Learning

TL;DR

This paper analyzes how the step size in gradient descent influences neural network training dynamics, convergence behavior, and practical training phenomena, using a dynamical systems perspective and Lyapunov stability analysis.

Contribution

It introduces a dynamical systems framework to understand the impact of step size on convergence and training outcomes in neural networks, explaining observed practical phenomena.

Findings

Step size determines the subset of local optima reachable during training.

Larger step sizes can lead to oscillations or convergence to different solutions.

The analysis explains phenomena like increased depth deterioration and residual network performance.

Abstract

Training a neural network with the gradient descent algorithm gives rise to a discrete-time nonlinear dynamical system. Consequently, behaviors that are typically observed in these systems emerge during training, such as convergence to an orbit but not to a fixed point or dependence of convergence on the initialization. Step size of the algorithm plays a critical role in these behaviors: it determines the subset of the local optima that the algorithm can converge to, and it specifies the magnitude of the oscillations if the algorithm converges to an orbit. To elucidate the effects of the step size on training of neural networks, we study the gradient descent algorithm as a discrete-time dynamical system, and by analyzing the Lyapunov stability of different solutions, we show the relationship between the step size of the algorithm and the solutions that can be obtained with this algorithm. The results provide an explanation for several phenomena observed in practice, including the deterioration in the training error with increased depth, the hardness of estimating linear mappings with large singular values, and the distinct performance of deep residual networks.