Understanding the Mechanisms of Fast Hyperparameter Transfer

The growing scale of deep learning models has rendered exhaustive hyperparameter (HP) optimization prohibitively expensive. A promising solution is the use of scale-aware HPs, which can enable direct transfer of optimal settings from small-scale grid searches to large models with minimal performance loss. Such approaches are useful when the optimal settings converge "fast" enough with scale. While approaches like the Maximal Update Parameterization ($\mu$P) have empirically displayed fast transfer when scaling model width, a deeper conceptual understanding of the mechanisms that enable this is still missing. Our work establishes a systematic conceptual framework for analyzing fast HP transfer across different synthetic and practical scenarios. In synthetic settings, we present various quantitative examples where transfer either offers a provable computational advantage or fails even under $\mu$P. We then propose a key property that enables the fast transfer often observed in practice: through a novel decomposition of the optimization trajectory, we identify one component that rapidly converges with model width and determines the optimal HPs, and the other that continues to improve the loss with increased width but has negligible impact on HP choice. We conjecture that this decomposition elucidates the key mechanisms behind fast transfer and empirically validate it in practical settings such as LLM training.