# Log-Normal Multiplicative Dynamics for Stable Low-Precision Training of Large Networks

arXiv: https://arxiv.org/abs/2506.17768 · HTML: https://mlsanity.com/paper/2506.17768/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 64 references that are in the map.
- First posted on arXiv 21 Jun 2025.

## Builds on (most cited first, 30 of 35)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Mixed Precision Training](https://mlsanity.com/paper/1710.03740/) (2017)
- [Up or Down? Adaptive Rounding for Post-Training Quantization](https://mlsanity.com/paper/2004.10568/) (2020)
- [signSGD: Compressed Optimisation for Non-Convex Problems](https://mlsanity.com/paper/1802.04434/) (2018)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [Variational Dropout and the Local Reparameterization Trick](https://mlsanity.com/paper/1506.02557/) (2015)
- [Symbolic Discovery of Optimization Algorithms](https://mlsanity.com/paper/2302.06675/) (2023)
- [Deep Learning with Limited Numerical Precision](https://mlsanity.com/paper/1502.02551/) (2015)
- [A Study of BFLOAT16 for Deep Learning Training](https://mlsanity.com/paper/1905.12322/) (2019)
- [FP8 Formats for Deep Learning](https://mlsanity.com/paper/2209.05433/) (2022)
- [Training Deep Neural Networks with 8-bit Floating Point Numbers](https://mlsanity.com/paper/1812.08011/) (2018)
- [Microscaling Data Formats for Deep Learning](https://mlsanity.com/paper/2310.10537/) (2023)
- [Flipout: Efficient Pseudo-Independent Weight Perturbations on Mini-Batches](https://mlsanity.com/paper/1803.04386/) (2018)
- [Fast and Scalable Bayesian Deep Learning by Weight-Perturbation in Adam](https://mlsanity.com/paper/1806.04854/) (2018)
- [Convolutional Neural Networks using Logarithmic Data Representation](https://mlsanity.com/paper/1603.01025/) (2016)
- [The Bayesian Learning Rule](https://mlsanity.com/paper/2107.04562/) (2021)
- [FP8-LM: Training FP8 Large Language Models](https://mlsanity.com/paper/2310.18313/) (2023)
- [Variational Learning is Effective for Large Deep Networks](https://mlsanity.com/paper/2402.17641/) (2024)
- [Optimizing Large Language Model Training Using FP4 Quantization](https://mlsanity.com/paper/2501.17116/) (2025)
- [With Shared Microexponents, A Little Shifting Goes a Long Way](https://mlsanity.com/paper/2302.08007/) (2023)
- [Anticorrelated Noise Injection for Improved Generalization](https://mlsanity.com/paper/2202.02831/) (2022)
- [Low-Pass Filtering SGD for Recovering Flat Optima in the Deep Learning Optimization Landscape](https://mlsanity.com/paper/2201.08025/) (2022)
- [Training LLMs with MXFP4](https://mlsanity.com/paper/2502.20586/) (2025)
- [Exponentiated Gradient Meets Gradient Descent](https://mlsanity.com/paper/1902.01903/) (2019)
- [Learning compositional functions via multiplicative weight updates](https://mlsanity.com/paper/2006.14560/) (2020)
- [Network Plasticity as Bayesian Inference](https://mlsanity.com/paper/1504.05143/) (2015)
- [Understanding Decoupled and Early Weight Decay](https://mlsanity.com/paper/2012.13841/) (2020)
- [Low-Precision Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2206.09909/) (2022)
- [Collage: Light-Weight Low-Precision Strategy for LLM Training](https://mlsanity.com/paper/2405.03637/) (2024)

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

- [M+Adam: Low-Precision Training via Additive-Multiplicative Optimization](https://mlsanity.com/paper/2607.10611/) (2026)
- [HORST: Composing Optimizer Geometries for Sparse Transformer Training](https://mlsanity.com/paper/2605.21104/) (2026)

## Abstract (arXiv)

Studies in neuroscience have shown that biological synapses follow a log-normal distribution whose transitioning can be explained by noisy multiplicative dynamics. Biological networks can function stably even under dynamically fluctuating conditions arising due to unreliable synaptic transmissions. Here we ask: Is it possible to design similar multiplicative training in artificial neural networks? To answer this question, we derive a Bayesian learning rule that assumes log-normal posterior distributions over weights which gives rise to a new Log-Normal Multiplicative Dynamics (LMD) algorithm. The algorithm uses multiplicative updates with both noise and regularization applied multiplicatively. The method is as easy to implement as Adam and only requires one additional vector to store. Our results show that LMD achieves stable and accurate training-from-scratch under low-precision forward operations for Vision Transformer and GPT-2. These results suggest that multiplicative dynamics, a biological feature, may enable stable low-precision inference and learning on future energy-efficient hardware.