Optimal Scaling Needs Optimal Norm
This provides a unifying principle for hyperparameter tuning in large-scale AI training, offering practical insights for researchers and practitioners, though it is incremental as it builds on existing scaling methods.
The paper tackles the problem of optimal hyperparameter scaling for large language models by discovering that joint optimal scaling across model and dataset sizes is governed by a single invariant: the operator norm of the output layer, termed norm transfer, with experiments on models up to 1.3B parameters and 138B tokens showing consistent norm values for optimal learning rate and batch size pairs.
Despite recent progress in optimal hyperparameter transfer under model and dataset scaling, no unifying explanatory principle has been established. Using the Scion optimizer, we discover that joint optimal scaling across model and dataset sizes is governed by a single invariant: the operator norm of the output layer. Across models with up to 1.3B parameters trained on up to 138B tokens, the optimal learning rate/batch size pair $(η^{\ast}, B^{\ast})$ consistently has the same operator norm value - a phenomenon we term norm transfer. This constant norm condition is necessary but not sufficient: while for each dataset size, multiple $(η, B)$ reach the optimal norm, only a unique $(η^{\ast}, B^{\ast})$ achieves the best loss. As a sufficient condition, we provide the first measurement of $(η^{\ast}, B^{\ast})$ scaling with dataset size for Scion, and find that the scaling rules are consistent with those of the Adam optimizer. Tuning per-layer-group learning rates also improves model performance, with the output layer being the most sensitive and hidden layers benefiting from lower learning rates. We provide practical insights on norm-guided optimal scaling and release our Distributed Scion (Disco) implementation with logs from over two thousand runs to support research on LLM training dynamics at scale.