Magnitude and Angle Dynamics in Training Single ReLU Neurons

TL;DR

This paper analyzes the dynamics of training single ReLU neurons by decomposing the process into magnitude and angle components, providing bounds and insights into convergence behavior, especially under small initialization, with experimental verification.

Contribution

It offers a theoretical framework for understanding the training dynamics of deep ReLU neurons through magnitude-angle decomposition, including bounds and convergence analysis.

Findings

Small scale initialization leads to slow convergence in deep ReLU neurons.

Theoretical bounds describe the evolution of magnitude and angle during training.

Experimental results verify the theoretical predictions.

Abstract

To understand learning the dynamics of deep ReLU networks, we investigate the dynamic system of gradient flow $w(t)$ by decomposing it to magnitude $w(t)$ and angle $\phi(t):= \pi - \theta(t) $ components. In particular, for multi-layer single ReLU neurons with spherically symmetric data distribution and the square loss function, we provide upper and lower bounds for magnitude and angle components to describe the dynamics of gradient flow. Using the obtained bounds, we conclude that small scale initialization induces slow convergence speed for deep single ReLU neurons. Finally, by exploiting the relation of gradient flow and gradient descent, we extend our results to the gradient descent approach. All theoretical results are verified by experiments.