# Omega-S: A Functional Resilience Index for LLM Fine-Tuning

arXiv: https://arxiv.org/abs/2608.03887 · HTML: https://mlsanity.com/paper/2608.03887/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 41 references that are in the map.
- First posted on arXiv 4 Aug 2026.

## Builds on (most cited first, 30 of 31)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [Learning without Forgetting](https://mlsanity.com/paper/1606.09282/) (2016)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [SmoothQuant: Accurate and Efficient Post-Training Quantization for Large Language Models](https://mlsanity.com/paper/2211.10438/) (2022)
- [SparseGPT: Massive Language Models Can Be Accurately Pruned in One-Shot](https://mlsanity.com/paper/2301.00774/) (2023)
- [LLM.int8(): 8-bit Matrix Multiplication for Transformers at Scale](https://mlsanity.com/paper/2208.07339/) (2022)
- [PyTorch FSDP: Experiences on Scaling Fully Sharded Data Parallel](https://mlsanity.com/paper/2304.11277/) (2023)
- [Orthogonal Gradient Descent for Continual Learning](https://mlsanity.com/paper/1910.07104/) (2019)
- [Orthogonal Subspace Learning for Language Model Continual Learning](https://mlsanity.com/paper/2310.14152/) (2023)
- [Outlier Weighed Layerwise Sparsity (OWL): A Missing Secret Sauce for Pruning LLMs to High Sparsity](https://mlsanity.com/paper/2310.05175/) (2023)
- [TRACE: A Comprehensive Benchmark for Continual Learning in Large Language Models](https://mlsanity.com/paper/2310.06762/) (2023)
- [Hutch++: Optimal Stochastic Trace Estimation](https://mlsanity.com/paper/2010.09649/) (2020)
- [AlphaPruning: Using Heavy-Tailed Self Regularization Theory for Improved Layer-wise Pruning of Large Language Models](https://mlsanity.com/paper/2410.10912/) (2024)
- [A topological insight into restricted Boltzmann machines](https://mlsanity.com/paper/1604.05978/) (2016)
- [On the Reproducibility of Neural Network Predictions](https://mlsanity.com/paper/2102.03349/) (2021)
- [Deterministic Implementations for Reproducibility in Deep Reinforcement Learning](https://mlsanity.com/paper/1809.05676/) (2018)
- [DarwinLM: Evolutionary Structured Pruning of Large Language Models](https://mlsanity.com/paper/2502.07780/) (2025)
- [Sculpting Subspaces: Constrained Full Fine-Tuning in LLMs for Continual Learning](https://mlsanity.com/paper/2504.07097/) (2025)
- [Mechanistic Analysis of Catastrophic Forgetting in Large Language Models During Continual Fine-tuning](https://mlsanity.com/paper/2601.18699/) (2026)
- [Non-Determinism in TensorFlow ResNets](https://mlsanity.com/paper/2001.11396/) (2020)
- [Fiedler Regularization: Learning Neural Networks with Graph Sparsity](https://mlsanity.com/paper/2003.00992/) (2020)
- [Efficient Shapley Value-based Non-Uniform Pruning of Large Language Models](https://mlsanity.com/paper/2505.01731/) (2025)
- [Muon-OGD: Muon-based Spectral Orthogonal Gradient Projection for LLM Continual Learning](https://mlsanity.com/paper/2605.08949/) (2026)
- [Spectral Manifold Regularization for Stable and Modular Routing in Deep MoE Architectures](https://mlsanity.com/paper/2601.03889/) (2026)
- [Subspace Geometry Governs Catastrophic Forgetting in Low-Rank Adaptation](https://mlsanity.com/paper/2603.02224/) (2026)
- [Catastrophic Forgetting is Low-Rank: A Function-Space Theory for Continual Adaptation](https://mlsanity.com/paper/2606.18024/) (2026)

## Abstract (arXiv)

Fine-tuning a large language model on new data degrades what it previously learned. We present Omega-S, a drop-in penalty computed from the weight matrix alone: it needs no previous-task data, no Fisher matrix and no stored copy of the old weights. It is three lines in an existing training loop and adds under 4% to the cost of a step. Retention. On Llama-3-8B with LoRA, fine-tuned from code to prose and measured by HumanEval over ten seeds, Omega-S retains more of the original capability than no regularisation on 9 of 10 seeds (0.173 -> 0.238 absolute pass@1; sign test one-sided p=0.011, Wilcoxon p=0.006), as a retention ratio, 62.9% -> 84.1%. It also beats tuned weight decay on 10 of 10 seeds (p=0.002) and tuned EWC on 8 of 10 (p=0.014), every arm re-measured in the same session. Mechanism, measured rather than asserted. Omega-S is topological by construction, its objective built from Tr(A^3), but we measured which of its four factors actually moves and three do not: their elasticity with respect to the weights is at or below 1e-4, against 9e-3 for the degree-variance term. As implemented, the composite reduces to a penalty on the variance of node degrees, which means row magnitude in square modules and directional alignment in non-square ones. We report this because a method whose name promises one thing and whose gradient does another should say so. We also enumerate the open design choices, including a contrast-preserving construction that does what it was designed to do and makes retention worse on all ten seeds. Repeating an identical configuration, same seed and same hardware, gives a standard deviation of 0.104 in retention ratio. We have not found this quantified for low-rank fine-tuning of language models, and it bounds every seed-paired comparison in this literature, ours included. Code, per-seed results and the full record of negative results are available.