O${}^2$PD: Overlap-Gated On-Policy Distillation on Reliable Prefix

On-policy distillation (OPD) supplies dense token-level supervision by aligning a student to the teacher's next-token distribution on its own generated rollouts. Existing works mostly implement top-$k$ OPD, applying reverse Kullback-Leibler ($RKL$) divergence between the teacher's and the student's renormalized distributions over teacher top-$k$ support. However, the teacher scores student-generated rollouts that are inherently off-policy for itself, leaving the reliability of its next-token guidance unclear. Restricting the divergence to the teacher top-$k$ support also discards the remaining mass outside that support. These gaps leave two questions implicit: 1) which tokens to align, and 2) how to align those tokens. To answer these questions, we propose Overlap-gated On-Policy Distillation (O${}^2$PD). As for 1), we quantitatively analyze the reliability of teacher supervision and find that alignment on the student's trajectory prefix is more reliable than alignment on the full context. Therefore, our O${}^2$PD is applied only to the 20% prefix tokens of each student rollout. As for 2), we theoretically characterize two problems of top-$k$ OPD: the Teacher-support Uncovering and the Dominant-Mode Mismatch. Based on these analyses, we propose two support strategies: the Reset Support}strategy attaches a reset atom to every token, pooling mass outside the teacher top-$k$, and the Support Convergence strategy converges the support to the already aligned intersection. Our O${}^2$PD employs the designed overlap-gated $\alpha$-divergence to interpolate between mode-covering and mode-seeking for per-token alignment, where the overlap is used as a gate to switch the support and adaptively modulate $\alpha$. Extensive experiments on seven common benchmarks demonstrate the superiority of O${}^2$PD, improving average performance by up to 5.29 points compared to OPD.