\boldsymbol{f}-OPD: Stabilizing Long-Horizon On-Policy Distillation with Freshness-Aware Control

On-policy distillation (OPD) trains a student on its own rollouts guided by teacher feedback and is becoming increasingly important for large language model (LLM) post-training. Like reinforcement learning (RL), however, OPD faces an on-policy systems bottleneck, as rollouts can dominate training time for reasoning workloads. Asynchronous training pipelines can alleviate this bottleneck by decoupling rollout generation from learner updates, but doing so introduces stale-policy data. While prior work has studied stale data in asynchronous RL, its effects in OPD remain underexplored. We present the first systematic study of staleness in asynchronous OPD, focusing on a practical setting where teacher feedback is implemented through local KL losses and full-vocabulary teacher logits are too expensive to store or transfer, necessitating finite teacher-score caches. We first show that KL direction changes the stale-data problem: teacher-weighted forward KL is more robust to stale rollouts, whereas student-weighted reverse KL is vulnerable. Second, for this vulnerable reverse-KL case, we study whether methods designed to stabilize asynchronous RL can mitigate OPD staleness. In our experiments, they do not improve over a simpler OPD-specific surrogate: recomputing the reverse-KL signal under the current student at learner time. Third, we analyze how finite teacher-score caches create a bias-variance tradeoff for sparse and sampled reverse-KL OPD estimators. This motivates multi-sample Monte Carlo (MC), which preserves MC correctability while reducing one-sample variance. Finally, we present and open-source AsyncOPD, a fully asynchronous OPD training pipeline built from these estimator choices. Experiments show that AsyncOPD improves training throughput by $1.6\times$ to $3.8\times$ over strict synchronous training while reaching comparable accuracy.

On-policy distillation (OPD) has emerged as an effective approach for large language model post-training, yet existing objectives face a trade-off between objective fidelity and optimization stability. Token-level OPD provides stable but local supervision, whereas sequence-level OPD captures future credit at the cost of horizon-dependent variance. We establish a unified temporal-credit view of these formulations, showing that practical token-level OPD can be interpreted as a temporal approximation to the sequence-level reverse-KL gradient. Building on this connection, we propose $γ$OPD, which uses discounted temporal credit assignment to balance long-horizon supervision and optimization stability, while admitting a horizon-independent variance bound. We further develop a reward-compatible bounded mixing (RBM) mechanism for $γ\mathrm{OPD}$ that balances verifiable outcome feedback with the discounted OPD advantage to move beyond purely teacher-dependent optimization. Experiments on mathematical and code reasoning demonstrate consistent improvements over existing OPD methods across vanilla, size-mismatched, and multi-teacher distillation settings.

On-Policy Distillation (OPD) facilitates the transfer of knowledge from domain expert to student in the post-training phase of Large Language Models (LLMs). However, the supervision signals in mainstream OPD methods suffer from high variance and noise which is generally instable during training. In this work, we systematically investigated what really matters to the performance and the fundamental mechanisms behind the instability during training. We found that retaining only the sign of token-level advantages is sufficient to achieve the performance comparable to standard OPD. Meanwhile, smoother and bounded advantages can stabilize the training process without sacrificing its performance. These motivated us to shape the advantages using the Total Variation (TV) and propose a robust TV regulated On-Policy Distillation (TV-OPD) method. Benefiting from the bounded and diminished advantages, TV-OPD exhibits stable training dynamics and steady late-stage performance. We conducted comprehensive experiments and found that, across various settings, TV-OPD consistently achieved better performance and lower variance in the late-stage of training.