# Learnable Multipliers: Freeing the Scale of Language Model Matrix Layers

arXiv: https://arxiv.org/abs/2601.04890 · HTML: https://mlsanity.com/paper/2601.04890/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 39 references that are in the map.
- First posted on arXiv 8 Jan 2026.

## Builds on (most cited first, 26 of 26)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [BLOOM: A 176B-Parameter Open-Access Multilingual Language Model](https://mlsanity.com/paper/2211.05100/) (2022)
- [GQA: Training Generalized Multi-Query Transformer Models from Multi-Head Checkpoints](https://mlsanity.com/paper/2305.13245/) (2023)
- [Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality](https://mlsanity.com/paper/2405.21060/) (2024)
- [Mixed Precision Training](https://mlsanity.com/paper/1710.03740/) (2017)
- [On Layer Normalization in the Transformer Architecture](https://mlsanity.com/paper/2002.04745/) (2020)
- [Few-Shot Parameter-Efficient Fine-Tuning is Better and Cheaper than In-Context Learning](https://mlsanity.com/paper/2205.05638/) (2022)
- [DoRA: Weight-Decomposed Low-Rank Adaptation](https://mlsanity.com/paper/2402.09353/) (2024)
- [MiniCPM: Unveiling the Potential of Small Language Models with Scalable Training Strategies](https://mlsanity.com/paper/2404.06395/) (2024)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Weight Normalization: A Simple Reparameterization to Accelerate Training of Deep Neural Networks](https://mlsanity.com/paper/1602.07868/) (2016)
- [Sharp Minima Can Generalize For Deep Nets](https://mlsanity.com/paper/1703.04933/) (2017)
- [ReZero is All You Need: Fast Convergence at Large Depth](https://mlsanity.com/paper/2003.04887/) (2020)
- [Cerebras-GPT: Open Compute-Optimal Language Models Trained on the Cerebras Wafer-Scale Cluster](https://mlsanity.com/paper/2304.03208/) (2023)
- [A Spectral Condition for Feature Learning](https://mlsanity.com/paper/2310.17813/) (2023)
- [Tensor Programs VI: Feature Learning in Infinite-Depth Neural Networks](https://mlsanity.com/paper/2310.02244/) (2023)
- [Don't be lazy: CompleteP enables compute-efficient deep transformers](https://mlsanity.com/paper/2505.01618/) (2025)
- [Why Do We Need Weight Decay in Modern Deep Learning?](https://mlsanity.com/paper/2310.04415/) (2023)
- [Falcon-H1: A Family of Hybrid-Head Language Models Redefining Efficiency and Performance](https://mlsanity.com/paper/2507.22448/) (2025)
- [Rotational Equilibrium: How Weight Decay Balances Learning Across Neural Networks](https://mlsanity.com/paper/2305.17212/) (2023)
- [Tensor Programs IVb: Adaptive Optimization in the Infinite-Width Limit](https://mlsanity.com/paper/2308.01814/) (2023)
- [Weight decay induces low-rank attention layers](https://mlsanity.com/paper/2410.23819/) (2024)
- [Weight Decay may matter more than muP for Learning Rate Transfer in Practice](https://mlsanity.com/paper/2510.19093/) (2025)
- [The Optimization Landscape of SGD Across the Feature Learning Strength](https://mlsanity.com/paper/2410.04642/) (2024)
- [Initialization of Large Language Models via Reparameterization to Mitigate Loss Spikes](https://mlsanity.com/paper/2410.05052/) (2024)

## Built on by (most cited first, 2 of 2)

- [Improving Neural Network Training by Decoupling the Magnitude and Direction of Weight Vectors](https://mlsanity.com/paper/2606.25971/) (2026)
- [A Mesoscopic View of Transformer Weights Through Row and Column Scale Fields](https://mlsanity.com/paper/2609.35852/) (2026)

## Abstract (arXiv)

Applying weight decay (WD) to matrix layers is standard practice in large-language-model pretraining. Prior work suggests that stochastic gradient noise induces a Brownian-like expansion of the weight matrices W, whose growth is counteracted by WD, leading to a WD-noise equilibrium with a certain weight norm ||W||. In this work, we view the equilibrium norm as a harmful artifact of the training procedure, and address it by introducing learnable multipliers to learn the optimal scale. First, we attach a learnable scalar multiplier to W and confirm that the WD-noise equilibrium norm is suboptimal: the learned scale adapts to data and improves performance. We then argue that individual row and column norms are similarly constrained, and free their scale by introducing learnable per-row and per-column multipliers. Our method can be viewed as a learnable, more expressive generalization of muP multipliers. It outperforms a well-tuned muP baseline, reduces the computational overhead of multiplier tuning, and surfaces practical questions such as forward-pass symmetries and the width-scaling of the learned multipliers. Finally, we validate learnable multipliers with both Adam and Muon optimizers, where it shows improvement in downstream evaluations matching the improvement of the switching from Adam to Muon.