Spherical Motion Dynamics: Learning Dynamics of Neural Network with Normalization, Weight Decay, and SGD

TL;DR

This paper investigates the learning dynamics of neural networks with normalization, weight decay, and SGD, revealing conditions for equilibrium and introducing angular update as a new measure, supported by theoretical proofs and experiments.

Contribution

It introduces assumptions leading to equilibrium in SMD, proves linear convergence of weight norm and angular update, and validates findings on large-scale vision tasks.

Findings

Weight norm converges at a linear rate under certain assumptions.

Angular update can be used as a measure and converges linearly.

Theoretical results align well with empirical observations on ImageNet and MSCOCO.

Abstract

In this work, we comprehensively reveal the learning dynamics of neural network with normalization, weight decay (WD), and SGD (with momentum), named as Spherical Motion Dynamics (SMD). Most related works study SMD by focusing on "effective learning rate" in "equilibrium" condition, where weight norm remains unchanged. However, their discussions on why equilibrium condition can be reached in SMD is either absent or less convincing. Our work investigates SMD by directly exploring the cause of equilibrium condition. Specifically, 1) we introduce the assumptions that can lead to equilibrium condition in SMD, and prove that weight norm can converge at linear rate with given assumptions; 2) we propose "angular update" as a substitute for effective learning rate to measure the evolving of neural network in SMD, and prove angular update can also converge to its theoretical value at linear rate; 3) we verify our assumptions and theoretical results on various computer vision tasks including ImageNet and MSCOCO with standard settings. Experiment results show our theoretical findings agree well with empirical observations.