Improving Neural Network Training by Decoupling the Magnitude and Direction of Weight Vectors
Authors: Alexander Hägele, Alejandro Hernández-Cano, Atli Kosson, Martin Jaggi
Organizations: Machine Learning and Optimization Lab, EPFL
Abstract
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.
The discovery of scaling laws has motivated training neural networks on ever increasing quantities of data. This is typically done with a constant decoupled weight decay which causes the network weights to shrink steadily over the course of training. Taking inspiration from the Robbins--Monro conditions, we propose to scale weight decay by the fraction of the peak learning rate η/ηmax. We prove that this scaled weight decay preserves the asymptotic stationarity guarantees of the corresponding unregularized methods for both stochastic gradient descent and the non-Euclidean spectral optimizer Muon, thereby avoiding the additional asymptotic bias introduced by constant decoupled weight decay. This retains the stability benefits of weight decay without changing the asymptotic optimization target. Using a steady-state analysis, we explain why under standard weight decay the weight norm shrinks steadily as training proceeds, whereas under scaled weight decay it settles to a roughly constant value. When applied to the training of mixture-of-experts models, Muon with scaled weight decay (Muon-SW) consistently outpaces Muon with identical hyperparameters, reaching the same validation loss 30% faster at our largest scale across models from 72−930 million parameters trained at ∼600 tokens per active parameter. If this trend continues to hold, the method promises to substantially accelerate the pre-training of frontier models while requiring only a few lines of code to implement.
Muon has emerged as a strong competitor to AdamW for language model pre-training, yet its behavior at scale is sensitive to weight decay. Recent work has observed that, for Muon without decoupled weight decay, the spectral norm of weight matrices drifts upward over training. Through a decomposition of the spectral norm into a row-magnitude factor and a row-coherence factor, we identify the former as the empirical driver of this drift under Muon, while the latter remains well-behaved along the trajectory. Motivated by this diagnosis, we introduce Muown, a drop-in replacement for Muon that treats the row-magnitude vector as an explicit optimizer variable, updating it under the ℓ∞ geometry induced by the decomposition, while applying Muon unchanged to the remaining direction component. We prove that Muown attains the optimal non-convex rates in both deterministic and stochastic regimes under a dual norm aligned with the underlying geometries and with a stochastic noise coefficient that empirically remains below that of Muon throughout training. Across GPT-style pre-training on FineWeb-Edu with model sizes from 124M up to 2.7B parameters, Muown improves perplexity over Muon, SOAP, AdamW, and Lion. It also widens the plateau of near-optimal learning rates across model scales, reduces sensitivity to weight decay, and avoids the spectral norm drift at negligible step-time overhead when appropriately sharded.
Many neural networks operations have a multiplicative nature rather than additive: halving or doubling a norm are analogous relatively but require unequal optimization distances when taking linear steps. Adaptive optimizers such as Adam normalize updates per coordinate, but update steps remain additive; weights with very different magnitudes receive similarly sized absolute changes, producing very different relative perturbations. We introduce \textbf{\method} (\textbf{\methodshort}), a weight reparameterization for neural networks that combines a sign-aware symmetric-exponential pathway with an identity-like linear pathway. The symmetric-exponential pathway is near-linear for small raw weights but increasingly curved at larger magnitudes. Additive updates in logarithmic space map to magnitude-proportional changes in effective weight space. The linear pathway provides a direct route through the transform that we hypothesize stabilizes optimization, while learnable scale, curvature, and offset parameters control balance between pathways and the curvature of the exponential pathway. These components create a curved parameter-space geometry that empirically improves speed of loss descent over standard linear parameterization. We also identify a useful \emph{mismatched initialization}: raw weights are chosen so a symmetric version of the transform matches Xavier statistics, but training uses an asymmetric forward transform that leaves positive weights at full strength while making negative weights smaller in magnitude; in small-model ablations, this improves early optimization and may act as a form of symmetry breaking. We train transformers on OpenWebText over nine width×depth configurations, \methodshort reaches matched validation loss in 1.32--1.49× fewer training steps, with the largest widths seeing the biggest gains.