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A Fisher-Geometric View of Multi-Teacher On-Policy Distillation

Liu Kaiyuan, Zhuang Ziyuan, Ma Haoxuan, Yannian Gu, Song Mingfei, Rongxiang Weng, Ye Jieping · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.26525 · cited by 0 works (OpenAlex) · open-access PDF

Chinese

再合并这个开放的领域专门基础模型到一个统一的总体主义者是后培训的核心目标。在复合中有两种方法是常用的:模型合并,它是快速的,但被处理为一个 heuristic,以及多教师在政策上的间接蒸,它匹配分布,但需要重大的聚合计算,并患有跨任务分层冲突的最小化,我们表明,这两种方法是由专业人士在不同的任务组合中看到的一个目标,所以在同一分类专家的共享池内,多教师在政策上的间接蒸的目标是,在第二个顺序上,正確地写了Fisher的分布不佳的目标。其最小化者是Fisher,一个经典的分层合并方法,由专门程序的密封式组合提供

Machine translation (Meta M2M100); the English original follows.

Abstract

Consolidating specialized foundation models into a unified generalist is a central goal of post-training. Two approaches are in common use: model merging, which is fast but treated as a heuristic, and multi-teacher on-policy distillation, which matches distributions but demands heavy rollout compute and suffers from cross-task gradient conflict. We show that the two are one objective seen at different orders. Within a shared basin of homologous specialists, the multi-teacher on-policy distillation objective is, to second order, exactly the Fisher-merging objective. Its minimizer is therefore the Fisher merge, a classic model-merging method given by a closed-form combination of the specialists, so merging is optimal, to second order, as a closed-form stage of distillation. Building on this identification, we propose Residual-Governed Branch-and-Merge(RGBM), a two-phase consolidation procedure. Concretely, a full-parameter Fisher merge is intractable on foundation models, so Phase~1 restricts the merge to the task-vector subspace. Extending tangent-space analyses of merging, we show that the same coefficients that mix task vectors also mix implicit rewards, which lets a residual program on unlabelled prompts solve for them. The program minimizes the worst per-domain residual and reports its energy-normalized values as the domain residual spectrum. Phase~2 lets this spectrum govern what may be added: RGBM passivates domains with low normalized residuals, trains isolated branches only for domains above the spectral gap, and writes back the admitted updates with calibrated shrinkage. On two open-source consolidation suites, the Phase~1 merge alone matches or outperforms fully trained multi-teacher distillation, and RGBM improves on it further with zero extra joint-training steps, whereas every un-gated write-back, including continued joint distillation from the merge, falls below it.

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Topics: Merge (version control), Residual, Distillation, Computer science, Mathematics, Algorithm

Sources: OpenAlex (CC0), arXiv, Crossref. Agent Exchange