Title: Distilling Reasoning into Student LLMs: Local Naturalness for Selecting Teacher Data
Primary area: generative models
Keywords: reasoning distillation, math, teacher, llm

Abstract: Distilling long reasoning traces (10K+ tokens) from stronger teacher models into
smaller student LLMs via supervised fine-tuning (SFT) has emerged as a standard
paradigm. This approach is both practical and efficient: it leverages the ease of
generating abundant reasoning data from stronger models and provides a direct,
data-driven way to teach less capable models better reasoning. While previous
work has largely focused on prompt selection with responses from a single teacher,
the equally important problem of choosing the best response when multiple teacher
outputs are available for a single prompt remains underexplored. This challenge
becomes especially important in a multi-teacher setting, where different students
may benefit from the outputs of different teachers. This paper fills that gap with a
systematic study of response selection for reasoning distillation. We first show that
the current method, which picks the response that the student assigns the highest
global log-probability (i.e., global "naturalness"), fails when responses come from
multiple teachers. In such cases, global naturalness no longer correlates with
downstream performance, especially as the reasoning traces from strong teachers
become longer. To overcome this limitation, we introduce Local Naturalness, which
scores a response by measuring the student’s log-probabilities over short, sequential
reasoning steps (e.g., sentences) conditioned only on a small local window. Local
Naturalness enables two novel applications: 1) Teacher Selection: Aggregating
local scores across prompts reliably identifies the most helpful teacher, whereas
global scoring fails completely. 2) Response Selection from a Mixed-Teacher
Dataset: When mixing answers from many teachers, Local Naturalness boosts a
32-billion-parameter student’s accuracy on math benchmarks by 9.4% over global-
naturalness-based selection, also surpassing the performance achieved by training
on data from the single best teacher. These results highlight the power of localized
data quality evaluation and data mixing for more effective reasoning distillation.