Extending μP: Spectral Conditions for Feature Learning Across Optimizers

Several variations of adaptive first-order and second-order optimization methods have been proposed to accelerate and scale the training of large language models. The performance of these optimization routines is highly sensitive to the choice of hyperparameters (HPs), which are computationally expensive to tune for large-scale models. Maximal update parameterization (μP) is a set of scaling rules which aims to make the optimal HPs independent of the model size, thereby allowing the HPs tuned on a smaller (computationally cheaper) model to be transferred to train a larger, target model. Despite promising results for SGD and Adam, deriving μP for other optimizers is challenging because the underlying tensor programming approach is difficult to grasp. Building on recent work that introduced spectral conditions as an alternative to tensor programs, we propose a novel framework to derive μP for a broader class of optimizers, including AdamW, ADOPT, LAMB, Sophia, Shampoo and Muon. We implement our μP derivations on multiple benchmark models and demonstrate zero-shot learning rate transfer across increasing model width for the above optimizers. Further, we provide empirical insights into depth-scaling parameterization for these optimizers.