DuoOPD: Learning from Joint Teacher-Student Outcomes for Multi-Task On-Policy Distillation

On-policy distillation (OPD) trains a student language model with dense feedback from a stronger teacher on student-generated trajectories. Yet standard OPD weights token-level distillation terms uniformly, implicitly treating local teacher preference as a proxy for correction utility. A decision's task value, however, depends on how the student completes the subsequent reasoning. This mismatch can cause imitation to suppress viable student strategies or reinforce paths the student cannot reliably execute. Verified trajectory outcomes provide complementary evidence about continuation quality, but do not directly identify the utility of individual decisions. We introduce Reward-Aligned Reweighting for On-Policy Distillation (R$^{2}$-OPD), which uses outcome agreement and the magnitude of teacher--student disagreement to continuously reallocate teacher supervision. It gives reward-aligned corrections greater relative influence while retaining dense feedback, moving beyond uniform imitation and hard filtering. Our analysis formalizes the mismatch between local teacher preference and student continuation value and establishes sufficient conditions for reallocation to improve first-order task progress over uniform OPD. Across seven mathematical reasoning benchmarks, R$^{2}$-OPD achieves the highest average accuracy among the compared training methods in both cross-size and same-size distillation. It outperforms standard OPD on all seven benchmarks, with average gains of 3.5 and 2.4 percentage points for 1.7B and 4B students, respectively. An extension to code generation yields an average gain of 1.6 percentage points over standard OPD. These results highlight outcome-guided supervision allocation as an effective way to translate dense teacher feedback into stronger student performance across model scales and task domains.

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) accelerates post-training by providing dense token-level supervision from a frozen teacher on the student's own rollouts. Vanilla OPD applies this supervision uniformly across prompts, without checking whether the teacher is reliable for each prompt. Because reverse KL is mode-seeking, a confidently wrong teacher can induce a strong yet misleading update. Distributional proxies, such as entropy or teacher-student likelihood agreement, measure uncertainty or agreement but do not directly verify outcome correctness. We introduce Teacher-Gated On-Policy Distillation (TGOPD), built on the principle that teacher reliability should be verified at the prompt level before dense supervision is admitted. TGOPD estimates reliability from a small set of verifier-scored teacher probes and routes each prompt exclusively to dense OPD when the reliability check passes or to verifier-grounded GRPO otherwise. Across 4B and 35B students in mathematics, code, and instruction following, TGOPD outperforms Vanilla OPD in all six single-domain settings and achieves higher seven-benchmark averages at both scales under multi-domain training. By using otherwise-idle teacher capacity for reliability estimation, TGOPD also reduces teacher-side compute waste in asynchronous OPD, increasing teacher-node GPU utilization from 9.8% to 78.9% in the measured 4B single-domain run.