Improving Neural Network Training by Decoupling the Magnitude and Direction of Weight Vectors
For practitioners training neural networks, this method simplifies hyperparameter tuning and improves training stability and performance across various optimizers and model scales.
The paper introduces Magnitude-Direction (MD) Decoupling, an optimizer modification that factorizes weight matrices into magnitude and direction components with separate learning rates, eliminating the need for weight decay and warmup. Across Adam and Muon optimizers, MD Decoupling improves on well-tuned baselines, transfers optimal learning rates across model widths without retuning, and benefits large Mixture-of-Experts models.
Modern neural network training relies on optimizers such as Adam and Muon which act on each weight matrix as a single object. Yet every weight matrix carries two distinct quantities -- a \emph{magnitude} and a \emph{direction} -- and all optimizers stepping in the matrix as a whole couple their dynamics: the directional change from an update depends on the current magnitude, while the magnitude drifts as a byproduct of learning the direction, so neither is governed directly by the learning rate. Typical training therefore leans on surrounding recipes such as weight decay and warmup to keep learning stable at scale, though these regulate the coupling only indirectly; other recent methods instead constrain the weight to a fixed-norm sphere, but add no learnable magnitude, leaving scale control to normalization layers alone. We propose \emph{Magnitude--Direction (MD) Decoupling}, an optimizer modification that factorizes each weight into a fixed-norm direction on a hypersphere and learnable per-row and per-column magnitude gains, updated at separate learning rates, all while the model still sees a single fused weight tensor. The method is agnostic to the base optimizer and removes the need for weight decay and warmup. Across both Adam and Muon, MD Decoupling improves on well-tuned baselines, transfers the optimal LR across model width without retuning, and continues to help at scale on large Mixture-of-Experts (MoE) models. Treating magnitude and direction as separately controlled quantities thus yields more predictable training dynamics and a simple, broadly applicable improvement to modern optimizers.