Rotational Equilibrium: How Weight Decay Balances Learning Across Neural Networks
This work provides a new perspective on training dynamics in deep learning, potentially improving optimization methods for researchers and practitioners, though it appears incremental in explaining existing techniques.
The study investigates how weight decay influences neuron updates in deep neural networks, revealing that it leads to rotational equilibrium states that balance learning rates across layers and neurons, and demonstrates that controlling rotation reduces the need for learning rate warmup.
This study investigates how weight decay affects the update behavior of individual neurons in deep neural networks through a combination of applied analysis and experimentation. Weight decay can cause the expected magnitude and angular updates of a neuron's weight vector to converge to a steady state we call rotational equilibrium. These states can be highly homogeneous, effectively balancing the average rotation -- a proxy for the effective learning rate -- across different layers and neurons. Our work analyzes these dynamics across optimizers like Adam, Lion, and SGD with momentum, offering a new simple perspective on training that elucidates the efficacy of widely used but poorly understood methods in deep learning. We demonstrate how balanced rotation plays a key role in the effectiveness of normalization like Weight Standardization, as well as that of AdamW over Adam with L2-regularization. Finally, we show that explicitly controlling the rotation provides the benefits of weight decay while substantially reducing the need for learning rate warmup.