Omega-S通过权重矩阵直接计算正则项，有效防止大模型微调时遗忘旧知识。
Omega-S: A Functional Resilience Index for LLM Fine-Tuning
- 仅用权重矩阵计算惩罚项，无需历史数据或额外存储
- 在10个种子中9次提升保留率，从62.9%升至84.1%
- 代码开源，结果可复现，适配低秩微调场景
在新数据上微调大语言模型会损害其先前学习能力。我们提出Omega-S，一种仅基于权重矩阵的即插即用正则项：无需前序任务数据、无需Fisher矩阵、也不需保存旧权重副本。只需三行代码嵌入训练循环，单步成本增加低于4%。在Llama-3-8B + LoRA设置下，从代码转向散文微调，以HumanEval衡量，相较于无正则化，在10个种子中有9个表现更优（绝对通过率从0.173提升至0.238；单边符号检验p=0.011，Wilcoxon p=0.006），保留率由62.9%升至84.1%。同时优于调参后的权重衰减（10/10胜出，p=0.002）和调参后的EWC（8/10胜出，p=0.014），所有对比均在同一会话中重新测量。机制实证：尽管构造上为拓扑性质，但实际仅有度方差项对权重敏感（弹性约9e-3），其余三项弹性≤1e-4。最终实现为节点度方差的惩罚，对应平方模块中的行幅值、非平方模块中的方向对齐。报告此现象，因名称承诺与梯度行为不一致者应明确说明。还列出开放设计选择，包括一种保持对比性的构造，虽符合设计初衷，却导致所有10个种子保留率下降。相同配置重复运行，保留率标准差为0.104，该数值此前未在语言模型低秩微调中量化，且为本领域所有种子配对比较提供上限。代码、每种子结果及完整负结果记录均已公开。
原文摘要 · Abstract (English)
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.
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