A Multi-Power Law for Loss Curve Prediction Across Learning Rate Schedules

Training large language models is both resource-intensive and time-consuming, making it crucial to understand the quantitative relationship between model performance and hyperparameters. In this paper, we derive an empirical law that is able to predict the entire pretraining loss curve across various learning rate schedules, including constant, cosine, and step decay schedules. This law takes a multi-power form, combining a power law based on the sum of learning rates and additional power laws to account for a loss reduction effect as learning rate decays. We validate this law extensively on Llama-2 models of varying sizes and demonstrate that, after fitting on a few learning rate schedules, it accurately predicts the loss curves for unseen schedules of different shapes and horizons. Moreover, by minimizing the predicted final pretraining loss across learning rate schedules, we are able to find a schedule that outperforms the widely-used cosine learning rate schedule. Interestingly, this automatically discovered schedule bears some resemblance to the recently proposed Warmup-Stable-Decay (WSD) schedule (Hu et al., 2024) but achieves a slightly lower final loss. We believe these results could offer valuable insights for understanding the dynamics of pretraining and for designing learning rate schedules to improve efficiency.