Beyond Verified Answers: Solver-Informed Self-Distillation for Bootstrapping Operations Research Language Models — Paper

Abstract

Modern large language models (LLMs) can translate natural-language descriptions into operations research (OR) formulations. Post-training techniques including reinforcement learning and on-policy self-distillation have further improved this capability. However, three limitations remain in training LLMs for OR formulations. First, training commonly relies on synthetic formulations validated by human experts or stronger models, constraining scalable supervision. Second, credit assignment is either coarse or costly: outcome rewards score an entire trajectory without locating the responsible modeling decision, whereas process-level supervision requires an additional evaluator. Third, privileged self-distillation can induce style mismatch by using solver context unavailable at deployment. We find that a model can improve from solver-artifact feedback generated by its own rollouts, making self-distillation a practical, evaluator-free source of dense supervision. Therefore, we propose SOLID: Solver-Informed On-Policy LearnIng through Self-Distillation, a novel framework for self-improving OR language models without verified answers or external evaluators. SOLID executes candidate programs from multiple rollouts, clusters their objectives, and selects a majority-group artifact as a pseudo-reference. The model then performs updates using group-relative advantages and dense self-supervision signals. Across multiple OR benchmarks, SOLID improves solution accuracy for both general-purpose and OR-tuned models over outcome-only group-relative training. These results show that solver artifacts can support scalable self-improvement without trusted answers.

Authors 5

Chenyu Zhou, Dongdong Ge, Jianghao Lin, Minglong Cao, Rui Zhu

Specification

Official page: arxiv.org/abs/2609.09957
Source:arXiv (Atom API + RSS)T1observed 2 h agohigh
Arxiv announce type: cross
Source:arXiv (Atom API + RSS)T1observed 2 h agohigh
arXiv id: 2609.09957
Source:arXiv (Atom API + RSS)T1observed 2 h agohigh
Categories: cs.AI, math.OC
Source:arXiv (Atom API + RSS)T1observed 2 h agohigh
PDF: arxiv.org/pdf/2609.09957
Source:arXiv (Atom API + RSS)T1observed 2 h agohigh
Primary category: math.OC
Source:arXiv (Atom API + RSS)T1observed 2 h agohigh
Published: 12 Sept 2026
Source:arXiv (Atom API + RSS)T1observed 2 h agohigh

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9

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T19

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2 h ago

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Authors: Rui Zhu, Minglong Cao, Chenyu Zhou

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9 claims · 9 properties

Official pageofficial_url1

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https://arxiv.org/abs/2609.09957	12 Sept 2026 04:12 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Abstractabstract1

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Modern large language models (LLMs) can translate natural-language descriptions into operations research (OR) formulations. Post-training techniques including reinforcement learning and on-policy self-distillation have further improved this capability. However, three limitations remain in training LLMs for OR formulations. First, training commonly relies on synthetic formulations validated by human experts or stronger models, constraining scalable supervision. Second, credit assignment is either coarse or costly: outcome rewards score an entire trajectory without locating the responsible modeling decision, whereas process-level supervision requires an additional evaluator. Third, privileged self-distillation can induce style mismatch by using solver context unavailable at deployment. We find that a model can improve from solver-artifact feedback generated by its own rollouts, making self-distillation a practical, evaluator-free source of dense supervision. Therefore, we propose SOLID: Solver-Informed On-Policy LearnIng through Self-Distillation, a novel framework for self-improving OR language models without verified answers or external evaluators. SOLID executes candidate programs from multiple rollouts, clusters their objectives, and selects a majority-group artifact as a pseudo-reference. The model then performs updates using group-relative advantages and dense self-supervision signals. Across multiple OR benchmarks, SOLID improves solution accuracy for both general-purpose and OR-tuned models over outcome-only group-relative training. These results show that solver artifacts can support scalable self-improvement without trusted answers.	12 Sept 2026 04:12 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Arxiv announce typearxiv_announce_type1

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cross	12 Sept 2026 04:12 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

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2609.09957	12 Sept 2026 04:12 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Authorsauthors1

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Chenyu Zhou, Dongdong Ge, Jianghao Lin, Minglong Cao, Rui Zhu	12 Sept 2026 04:12 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Categoriescategories1

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cs.AI, math.OC	12 Sept 2026 04:12 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

PDFpdf_url1

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https://arxiv.org/pdf/2609.09957	12 Sept 2026 04:12 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

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math.OC	12 Sept 2026 04:12 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

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12 Sept 2026	12 Sept 2026 04:12 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

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12 Sept 2026
New paperPaperBeyond Verified Answers: Solver-Informed Self-Distillation for Bootstrapping Operations Research Language Models
New paper: Beyond Verified Answers: Solver-Informed Self-Distillation for Bootstrapping Operations Research Language Models
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Source documents
Source	Document	Type	Tier	Last observed	Snapshots
arXiv (Atom API + RSS)	rss.arxiv.org/rss/cs.AI	feed	T1· Official	2 h ago	2

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