Theory of Optimal Learning Rate Schedules and Scaling Laws for a Random Feature Model
arXiv:2602.04774v1 Announce Type: cross
Abstract: Setting the learning rate for a deep learning model is a critical part of successful training, yet choosing this hyperparameter is often done empirically with trial and error. In this work, we explore a solvable model of optimal learning rate schedules for a powerlaw random feature model trained with stochastic gradient descent (SGD). We consider the optimal schedule $eta_T^star(t)$ where $t$ is the current iterate and $T$ is the total training horizon. This schedule is computed both numerically and analytically (when possible) using optimal control methods. Our analysis reveals two regimes which we term the easy phase and hard phase. In the easy phase the optimal schedule is a polynomial decay $eta_T^star(t) simeq T^{-xi} (1-t/T)^{delta}$ where $xi$ and $delta$ depend on the properties of the features and task. In the hard phase, the optimal schedule resembles warmup-stable-decay with constant (in $T$) initial learning rate and annealing performed over a vanishing (in $T$) fraction of training steps. We investigate joint optimization of learning rate and batch size, identifying a degenerate optimality condition. Our model also predicts the compute-optimal scaling laws (where model size and training steps are chosen optimally) in both easy and hard regimes. Going beyond SGD, we consider optimal schedules for the momentum $beta(t)$, where speedups in the hard phase are possible. We compare our optimal schedule to various benchmarks in our task including (1) optimal constant learning rates $eta_T(t) sim T^{-xi}$ (2) optimal power laws $eta_T(t) sim T^{-xi} t^{-chi}$, finding that our schedule achieves better rates than either of these. Our theory suggests that learning rate transfer across training horizon depends on the structure of the model and task. We explore these ideas in simple experimental pretraining setups.
