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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

German

Die Konsolidierung dieses spezialisierten Domain-Grundmodells in ein vereinheitlichtes Generalist ist ein zentrales Ziel des Post-Training-Extraining. Zwei Ansätze sind in gemeinsamer Verwendung: Modell-Fusion, das schnell, aber als heuristisch behandelt wird, und die multi-teacher-out-policy-specidual-Destillation, die Verteilungen entspricht, aber ein schweres Rollover-Computing-Ziel erfordert. Sein Minimierer kann daher der Fisher-Fusion sein, eine klassische Branchen-Gradient-Konfliktkonflikt. Wir zeigen, dass die beiden ein Ziel sind, das von einer geschlossenen Aufgabenkombination der Spezialisten gesehen wird. In einem gemeinsamen Becken von homologen Spezialisten ist das Multi-teacher-out-policy-

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

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