# Understanding LLM Parameter Update Sparsity through the Lens of Fisher

arXiv: https://arxiv.org/abs/2609.36262 · HTML: https://mlsanity.com/paper/2609.36262/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 24 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 22 of 22)

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- [On-Policy Distillation of Language Models: Learning from Self-Generated Mistakes](https://mlsanity.com/paper/2306.13649/) (2023)
- [Transformer Feed-Forward Layers Build Predictions by Promoting Concepts in the Vocabulary Space](https://mlsanity.com/paper/2203.14680/) (2022)
- [New insights and perspectives on the natural gradient method](https://mlsanity.com/paper/1412.1193/) (2014)
- [Training Neural Networks with Fixed Sparse Masks](https://mlsanity.com/paper/2111.09839/) (2021)
- [RL's Razor: Why Online Reinforcement Learning Forgets Less](https://mlsanity.com/paper/2509.04259/) (2025)
- [Raise a Child in Large Language Model: Towards Effective and Generalizable Fine-tuning](https://mlsanity.com/paper/2109.05687/) (2021)
- [Universal Neurons in GPT2 Language Models](https://mlsanity.com/paper/2401.12181/) (2024)
- [The Path Not Taken: RLVR Provably Learns Off the Principals](https://mlsanity.com/paper/2511.08567/) (2025)
- [Reinforcement Learning Finetunes Small Subnetworks in Large Language Models](https://mlsanity.com/paper/2505.11711/) (2025)
- [Dense Supervision, Sparse Updates: On the Sparsity and Geometry of On-Policy Distillation](https://mlsanity.com/paper/2606.13657/) (2026)
- [Understanding and Exploiting Weight Update Sparsity for Communication-Efficient Distributed RL](https://mlsanity.com/paper/2602.03839/) (2026)
- [Do We Need Adam? Surprisingly Strong and Sparse Reinforcement Learning with SGD in LLMs](https://mlsanity.com/paper/2602.07729/) (2026)
- [Sparsity May Be All You Need: Sparse Random Parameter Adaptation](https://mlsanity.com/paper/2502.15975/) (2025)
- [The Multiple Ticket Hypothesis: Random Sparse Subnetworks Suffice for RLVR](https://mlsanity.com/paper/2602.01599/) (2026)
- [RL Forgets! Towards Continual Policy Optimization](https://mlsanity.com/paper/2607.04364/) (2026)
- [Information Guided Regularization for Fine-tuning Language Models](https://mlsanity.com/paper/2406.14005/) (2024)
- [Information-Geometric Forward Policy Training in GFlowNets](https://mlsanity.com/paper/2608.03967/) (2026)

## Abstract (arXiv)

Recent studies have observed that parameter changes during language-model post-training can be concentrated in a small subset of coordinates. This phenomenon has been reported in reinforcement learning, on-policy distillation, and supervised fine-tuning on near-policy data. Its recurrence across different post-training paradigms suggests shared structure in training dynamics. In this paper, we examine this pattern through the diagonal model Fisher, which measures the sensitivity of the model's output distribution to individual parameters and is independent of any particular reward or teacher signal. Theoretically, we show that small diagonal Fisher leads to small expected gradients across a range of training objectives, providing a common explanation for sparse gradient updates. Empirically, we test this connection in RL and OPD. We find that Fisher identifies where gradients are concentrated, and fixed sparse masks selected from the initial Fisher retain a large proportion of the improvement from full training. Finally, we investigate the mechanisms underlying low Fisher in on-policy training. Our results show that high-probability next tokens tend to have similar parameter sensitivities, contributing to low Fisher. Together, these results establish the diagonal model Fisher as a unifying perspective linking update sparsity to on-policy training dynamics in LLM post-training.