Towards joint scaling laws with optimal batch size schedules

We study the scaling behavior of learning rate and batch size in pretraining dense large language models on English-prevalent corpora. Beyond scaling jointly optimal learning rates and batch sizes, we investigate their marginal evolution with model capacity and data scale and develop a model that captures these relationships. As we employ a Warmup-Stable-Decay learning rate schedule, we further investigate the gains from learning rate annealing over a broad range of hyperparameters settings, models and data budgets, and whether the optimal learning rate and batch size transfer between the stable and decay phases. Finally, we characterize the dependence of loss on model capacity and dataset size, evaluating recently proposed scaling forms that explicitly model their interaction. We find these approaches particularly effective at capturing both undertraining and overtraining regimes across our experiments. This study establishes a first baseline and scaling procedure for the development of future OpenEuroLLM models. We open-source the complete collection of pretraining runs used in this study.

Schedule-Free Learning has shown promise as a practical anytime training method for machine learning, showing success across dozens of standard benchmark problems. However, strong performance for LLM training has only been demonstrated at small scales. We identify a number of fixes necessary to scale up Schedule-Free Learning to larger batch sizes and model sizes, and present a learning-rate-free and schedule-free method (ScheduleFree+) for training large language models which greatly outperforms Warmup-Stable-Decay (WSD) schedules. We also demonstrate that Schedule-Free Learning is most effective for long duration training, and at 1000 tokens per parameter, it outperforms SOTA schedules by 31%. Schedule-Free Learning provides a theoretical foundation for the use of model averaging and checkpoint merging during pretraining.

Learning-rate transfer can reduce the cost of training large language models: instead of sweeping learning rates at target scale, practitioners extrapolate from smaller runs. Existing approaches often assume that the optimal learning rate follows a log-linear scaling law in data scale and model size. We carefully examine and evaluate this scaling law. In our empirical study of GPT-2--style models from 22M to 707M parameters trained on 5B to 100B tokens, the optimal learning rate develops upward curvature at larger scales, leading to inaccurate extrapolation. We find that this curvature largely disappears when learning rates are replaced by effective learning rate (the step size in normalized weight space), and when data $D$ extrapolation is used instead of model size $N$ extrapolation. Next, we explain nonlinearity in scaling: weight-norm converges to equilibrium slower when optimal learning is small, requiring a larger step size to reduce the transient phase. Experiments with AdamH, which directly controls the effective learning rate, further support this explanation.