μpscaling small models: Principled warm starts and hyperparameter transfer

Hyperparameter transfer allows extrapolating optimal optimization hyperparameters from small to large scales, making it critical for training large language models (LLMs). This is done either by fitting a scaling law to the hyperparameters or by a judicious choice of parameterization, such as Maximal Update ($μ$P), that renders optimal hyperparameters approximately scale invariant. In this paper, we first develop a framework to quantify hyperparameter transfer through three metrics: (1) the quality of the scaling law fit, (2) the robustness to extrapolation errors, and (3) the asymptotic loss penalty due to choice of parameterization. Next, we investigate through a comprehensive series of ablations why $μ$P appears to offer high-quality learning rate transfer relative to standard parameterization (SP), as existing theory is inadequate. We find that the overwhelming benefit of $μ$P relative to SP when training with AdamW arises simply from maximizing the learning rate of the embedding layer. In SP, the embedding layer learning rate acts as a bottleneck that induces training instabilities; increasing it by a factor of width to match $μ$P dramatically smooths out training while improving hyperparameter transfer. We also find that weight decay improves the scaling law fits, while, in the fixed token-per-parameter setting, it hurts the robustness of the extrapolation.

Scaling laws promised cost-effective experiments; six years later, they have yet to fully deliver. Instead, researchers have found them unreliable at small scales (starting at 4M parameters) and concluded that sizable models cannot be avoided. We show this is not the case: the confounding factor is hyperparameters. Small models are highly sensitive, but hyperparameter sensitivity fades with scale. This small-scale sensitivity makes scaling laws easy to miss because they only emerge on the fully tuned frontier, and reaching that frontier requires an extensive search far beyond what most ever run. By ablating the basic scaling law recipe, we show well-tuned hyperparameters matter more than any other ingredient. Further, we reveal why those hyperparameters become easier to find: as scale increases, the hyperparameter loss surface becomes lower dimensional. Nevertheless while scaling laws exist in small models, extrapolation hits statistical limitations. A holistic approach is required. Synthesizing our insights with the recent literature, we develop a new methodology for model-centric research and demonstrate it on a question that once took the field years to settle: where to place normalization layers in the transformer architecture. From small-scale experiments, we recover the large scale result: pre-normalization works better as models grow in size. With the right tools and a better understanding, small-scale experiments can deliver on scaling laws' long-awaited promise.

We present a functional form (that we refer to as a Unified Neural Scaling Law (UNSL)) that accurately models and extrapolates the scaling behaviors of deep neural networks as multiple dimensions all vary simultaneously (i.e. how the evaluation metric of interest varies as one simultaneously varies the number of model parameters, training dataset size, number of training steps, number of inference steps, amount of compute, and various hyperparameters) for various architectures and for each of various tasks within a varied set of upstream and downstream tasks. This set includes large-scale vision, language, math, and reinforcement learning. When compared to other functional forms for neural scaling, this functional form yields extrapolations of scaling behavior that are considerably more accurate on this set.