Spherical Motion Dynamics: Learning Dynamics of Neural Network with Normalization, Weight Decay, and SGD
This work provides a theoretical foundation for SMD, addressing a gap in explaining equilibrium conditions, which is incremental but important for researchers in deep learning optimization.
The paper tackles the problem of understanding the learning dynamics of neural networks with normalization, weight decay, and SGD, known as Spherical Motion Dynamics (SMD), by investigating the cause of the equilibrium condition and proving convergence rates for weight norm and angular update. The results show theoretical findings align with empirical observations on tasks like ImageNet and MSCOCO.
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.