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Large Language Models (LLMs) have performed well on various reasoning tasks,but their inaccessibility and numerous parameters hinder wide application inpractice. One promising way is distilling the reasoning ability from LLMs tosmall models by the generated chain-of-thought reasoning paths. In some cases,however, LLMs may produce incorrect reasoning chains, especially when facingcomplex mathematical problems. Previous studies only transfer knowledge frompositive samples and drop the synthesized data with wrong answers. In thiswork, we illustrate the merit of negative data and propose a modelspecialization framework to distill LLMs with negative samples besides positiveones. The framework consists of three progressive steps, covering from trainingto inference stages, to absorb knowledge from negative data. We conductextensive experiments across arithmetic reasoning tasks to demonstrate the roleof negative data in distillation from LLM.

Turning Dust into Gold: Distilling Complex Reasoning Capabilities from LLMs by Leveraging Negative Data