Beyond GRPO and On-Policy Distillation: An Empirical Sparse-to-Dense Reward Principle for Language-Model Post-Training

In settings where labeled verifiable training data is the binding constraint, each checked example should be allocated to the model and reward density where it is most informative. We identify a reward-density principle: sparse sequence-level reward is most useful on models that can explore and discover better behavior, while dense token-level teacher supervision is better suited for compressing that behavior into a smaller deployment model; accordingly, scarce labeled data should be used upstream on the strongest available teacher, then transferred downstream as dense supervision. We evaluate this rule through a four-stage workflow:teacher RL, forward-KL warmup, on-policy distillation, optional post-bridge student RL on verifiable math and code generation with Qwen3, Llama, and Qwen2.5 models. At fixed Qwen3-1.7B deployment-student size, an RL-improved 8B teacher distilled through the dense bridge outperforms direct GRPO on the same student (79.3% vs. 75.9% on MATH; 25.2% vs. 19.8% on AIME~2024), while transfer from the same teacher before RL underperforms. A component ablation confirms that each stage is load-bearing: the RL-improved teacher adds 7.8 absolute points on MATH over a raw teacher of the same size, the forward-KL warmup adds 1.7, and on-policy distillation adds 3.3. The teacher-quality ordering: raw-teacher transfer < direct GRPO < RL-teacher transfer, replicates on Llama-3.1-8B-Instruct with a Llama-3.3-70B-Instruct teacher and on Qwen2.5-7B-Instruct for code generation with a Qwen2.5-32B-Instruct teacher. The operational lesson is to avoid spending scarce labeled data on the least prepared policy: use sparse reward for teacher-side discovery, dense transfer for student compression, and student-side sparse reward only after the bridge. The OPD implementation used in our experiments is available at https://github.com/HJSang/OPSDOnPolicyDistillation.