How to Allocate Your Tokens? Scaling Laws with Training Steps and Batch Size

We propose a scaling law that takes into account model size and training data while explicitly splitting the latter into training steps and batch size (called *three-term law*). Fitting the proposed law on a large set of training runs, we find that it correctly recovers the scaling of the optimal batch size. Making use of training runs with suboptimal batch size, our proposed law can be robustly fit with a significantly smaller amount of training runs. We further show that the three-term law can be used to derive scaling laws for suboptimal batch sizes, and that it matches previous empirical findings related to the critical batch size.