Stepping on the Edge: Curvature Aware Learning Rate Tuners

Abstract

Curvature information -- particularly, the largest eigenvalue of the loss Hessian, known as the sharpness -- often forms the basis for learning rate tuners. However, recent work has shown that the curvature information undergoes complex dynamics during training, going from a phase of increasing sharpness to eventual stabilization. We analyze the closed-loop feedback effect between learning rate tuning and curvature. We find that classical learning rate tuners may yield greater one-step loss reduction, yet they ultimately underperform in the long term when compared to constant learning rates in the full batch regime. These models break the stabilization of the sharpness, which we explain using a simplified model of the joint dynamics of the learning rate and the curvature. To further investigate these effects, we introduce a new learning rate tuning method, Curvature Dynamics Aware Tuning (CDAT), which prioritizes long term curvature stabilization over instantaneous progress on the objective. In the full batch regime, CDAT shows behavior akin to prefixed warm-up schedules on deep learning objectives, outperforming tuned constant learning rates. In the mini batch regime, we observe that stochasticity introduces confounding effects that explain the previous success of some learning rate tuners at appropriate batch sizes. Our findings highlight the critical role of understanding the joint dynamics of the learning rate and curvature, beyond greedy minimization, to diagnose failures and design effective adaptive learning rate tuners.