Beyond Rejection Sampling: Trajectory Fusion for Scaling Mathematical Reasoning

TL;DR

TrajFusion is a novel fine-tuning method for large language models that models trial-and-error reasoning by fusing correct and incorrect trajectories, improving performance on complex mathematical reasoning tasks.

Contribution

It introduces TrajFusion, a structured supervision construction approach that enhances mathematical reasoning in LLMs without changing architecture or objectives.

Findings

TrajFusion outperforms rejection sampling fine-tuning on math benchmarks.

It improves reasoning on challenging and long-form problems.

The method adaptively controls trajectory length based on error signals.

Abstract

Large language models (LLMs) have made impressive strides in mathematical reasoning, often fine-tuned using rejection sampling that retains only correct reasoning trajectories. While effective, this paradigm treats supervision as a binary filter that systematically excludes teacher-generated errors, leaving a gap in how reasoning failures are modeled during training. In this paper, we propose TrajFusion, a fine-tuning strategy that reframes rejection sampling as a structured supervision construction process. Specifically, TrajFusion forms fused trajectories that explicitly model trial-and-error reasoning by interleaving selected incorrect trajectories with reflection prompts and correct trajectories. The length of each fused sample is adaptively controlled based on the frequency and diversity of teacher errors, providing richer supervision for challenging problems while safely reducing to vanilla rejection sampling fine-tuning (RFT) when error signals are uninformative. TrajFusion requires no changes to the architecture or training objective. Extensive experiments across multiple math benchmarks demonstrate that TrajFusion consistently outperforms RFT, particularly on challenging and long-form reasoning problems.