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原文摘要

Training smaller language models on reasoning solutions generated by a larger teacher model is the prevailing approach for transferring mathematical reasoning capabilities. However, the student tends to overfit the output distribution of the teacher and generalizes poorly when its own behavior at inference time deviates from the training data. Recent methods attempt to bridge this gap by incorporating error data, yet the corrective trajectories still reflect the reasoning patterns of the teacher, which the student cannot reliably reproduce, leaving the distribution mismatch largely unresolved. We propose a method, Failure Profiling and Reachable Trajectory Selection (FPRT), built on the insight that effective training data should be jointly determined by what the student fails on and what it can realistically learn. Our method first profiles the failure behavior of the student on each problem through repeated sampling, characterizing it in terms of error rate and error consistency to reveal qualitatively distinct failure modes. Each mode demands a fundamentally different training signal. The diagnosed mode then guides candidate generation from the teacher via distinct prompting strategies tailored to each failure type. Among the correct candidates, our method selects the one whose initial reasoning steps after the point of divergence are most reproducible by the student. These first corrective steps constitute the primary bottleneck, while the remaining steps follow an already corrected context and can therefore be continued more readily. This contrasts with standard reranking approaches, which score the trajectory as a whole and thereby dilute the signal at the critical reasoning transition. Experiments demonstrate that our method achieves the SOTA performance across the five benchmarks on both student backbones.