A Physics-Inspired Optimizer: Velocity Regularized Adam

We introduce Velocity-Regularized Adam (VRAdam), a physics-inspired optimizer for training deep neural networks that draws on ideas from quartic terms for kinetic energy with its stabilizing effects on various system dynamics. Previous algorithms, including the ubiquitous Adam, operate at the so called adaptive edge of stability regime during training leading to rapid oscillations and slowed convergence of loss. However, VRAdam adds a higher order penalty on the learning rate based on the velocity such that the algorithm automatically slows down whenever weight updates become large. In practice, we observe that the effective dynamic learning rate shrinks in high-velocity regimes, damping oscillations and allowing for a more aggressive base step size when necessary without divergence. By combining this velocity-based regularizer for global damping with per-parameter scaling of Adam to create a hybrid optimizer, we demonstrate that VRAdam consistently exceeds the performance against standard optimizers including AdamW. We benchmark various tasks such as image classification, language modeling, image generation and generative modeling using diverse architectures and training methodologies including Convolutional Neural Networks (CNNs), Transformers, and GFlowNets.