用李括号预测学习顺序，让多阶段训练更高效。
The Geometry of Sequential Learning: Lie-Bracket Prediction of Transfer Order
- 用梯度更新场的李括号计算学习顺序的局部影响。
- 在指令微调和DPO任务中，配对准确率达98.1%~98.9%。
- 适合需要优化多个预训练源顺序的大型语言模型训练者。
顺序学习具有依赖性：从Pile风格的下一个词域适应到指令SFT与DPO，N个候选源可产生N!种课程顺序。我们发现局部顺序效应由一个可计算的几何量——梯度更新场的李括号交换子决定，可生成一对比较得分，判断A→B或B→A在目标域上哪个更优。该成对括号原语还定义了李括号锦标赛：以共享的θ₀为目标梯度参考，利用海森对称性，仅需每个源一次海森-向量乘积，完成O(N)次点积和O(N log N)排序，无需显式构建O(N²)边矩阵。实验表明，该规划器在指令SFT/DPO任务中k=1时达到98.1%/98.9%的配对准确率，k=20时仍保持73.1%/72.2%，并在四个LLM上保留原始预训练域证据，准确率达82.4–92.0%，扩散模型上达91.1%。在课程规模下，87.5%的试验中恢复出所有3!种调度中的最优方案，在56个MMLU主题中达到99.0–99.6%采样百分位，显著优于报告的下降梯度范数基线。这些结果将顺序学习重新理解为几何锦标赛问题：李括号既提供局部配对顺序信息，又为多领域调度提供可扩展原语。
原文摘要 · Abstract (English)
Sequential learning is order-dependent: from Pile-style next-token domain adaptation to instruction-SFT and DPO, N candidate sources induce N! possible curricula. We show that the local order effect is governed by a computable geometric quantity, the Lie-bracket commutator of gradient update fields, yielding a pairwise score for whether A->B or B->A is better for a target domain. The pairwise bracket primitive also defines a Lie-Bracket Tournament: with a shared theta_0 target-gradient reference, Hessian symmetry gives Borda/row-sum scores from one Hessian-vector product per source, O(N) dot products, and an O(N log N) sort, without materializing the O(N^2) edge matrix. Empirically, the planner reaches 98.1%/98.9% pairwise accuracy at k=1 for instruction-SFT/DPO, remains at 73.1%/72.2% at k=20, and preserves the original pretraining-domain evidence with 82.4-92.0% accuracy across four LLMs and 91.1% on diffusion. At curriculum scale, it recovers the best of all 3! schedules in 87.5% of trials, ranks 85 Stack programming-language source domains for a Python target in the 99th sampled percentile, and reaches the 99.0-99.6th sampled percentile on 56 MMLU subjects, sharply above the reported descending gradient-norm baseline. These results reframe sequential learning as a geometric tournament problem: commutators provide both local pairwise order information and a scalable primitive for many-domain schedules.
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