$D^2$-OPD: Separating Direction and Distance in Extrapolative On-Policy Distillation

On-policy distillation (OPD) trains a student on its own generated responses using dense, token-level supervision from a stronger teacher. Vanilla OPD treats all teacher signals equally, assuming that the teacher's supervision is equally important for every token. However, teacher signals at different tokens may have very different effects on the student's performance: some correct important reasoning errors, while others have little effect on the final answer. Motivated by this observation, we introduce Dr. OPD (OPD Done Right), which defines the optimal weighted OPD to maximize the student's performance. We formulate Dr. OPD as a bilevel optimization problem in which the student learns from weighted teacher supervision, while the weights are selected to maximize the expected reward of the resulting student. To solve Dr. OPD, we develop an efficient iterative solver that updates the token weights and student policy alternatively. At each round, it updates weights in closed form and then takes one gradient step on the resulting weighted OPD objective. Under regularity conditions, we show that this weighted update achieves a higher expected reward than a vanilla OPD update. Empirically, across strong-to-weak and same-size distillation on math and code, Dr. OPD consistently outperforms all evaluated baselines. In particular, in the strong-to-weak distillation setting, Dr. OPD improves average math performance by $9.7$ points over vanilla OPD, and enables the smaller student to surpass its larger teacher.

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 $α$-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 $α$.

On-policy distillation (OPD) is a widely used technique to transfer capabilities from capable teacher language models to the base student models, and can be formulated in a reinforcement learning style objective using student generated rollouts. Yet, despite the divergence reward being dependent on student model likelihood, existing works usually adopt a stop gradient design primarily for stability, which makes the resulting advantage estimation questionable. In this work, we provide a generic optimization framework based on f-divergence between the student and teacher, and mathematically revisit whether such design space is valid. We prove that general stop-gradient operation would lead to biased estimates of the reward objective and corresponding gradient for general divergence functions. We propose OPD+, the corrected version of OPD that demonstrates improved performance over the baseline KL approach and also supports the choice of various f-divergence. We validate our findings on mathematical reasoning and tool-use benchmarks.