Meet Students Where They Are: Student-Prefix Teacher Adaptation for Reliable On-Policy Distillation

On-policy distillation (OPD) mitigates the student's exposure bias by optimizing on student-generated trajectories. However, student-generated prefixes may contain reasoning errors or follow patterns unfamiliar to the teacher, undermining the reliability of its token-level supervision. This degradation becomes more pronounced as prefixes grow longer. To address this teacher-side distribution mismatch, we propose **Student-Prefix Teacher Adaptation** (SPTA), which adapts the teacher to the states on which it is queried during distillation. Specifically, before OPD starts, we collect student rollout prefixes with random lengths and continually train the teacher to continue from the prefixes using reinforcement learning with verifiable rewards (RLVR). By treating each prefix as context and rewarding correct final answers, the teacher learns to recover from student-induced states rather than imitate the student's reasoning. Then, the adapted teacher is used for standard OPD, with no changes to the distillation objective. Experiments on mathematical reasoning benchmarks show that our SPTA improves average student performance over standard OPD and OPD with task-level RLVR-adapted teachers. These results highlight the importance of adapting teachers not only to solve problems independently, but also to provide reliable guidance on the states their students actually visit.