CoRD: Coupled Rollouts Fill the Blind Spot of On-Policy Distillation

On-policy distillation trains a student on its own rollouts, so states the teacher would reach but the student does not are seldom supervised. We introduce CoRD (coupled-rollout distillation), which keeps the on-policy loss and moves only the distribution of training states toward the teacher, drawing rollouts from a geometric interpolation of student and teacher through the maximal coupling of speculative decoding. The coupling keeps the student's token wherever the two distributions agree and otherwise substitutes a token from the part of the target the student underweights, so interventions are as rare as any exact sampler allows and recede as the student learns. Across six base students from two model families (Qwen3 and Qwen3.5, 0.6B to 4B), CoRD raises average pass@1 over on-policy distillation by 0.9 to 2.0 points and pass@8 by 1.2 to 3.4 points on five public math benchmarks at equal steps and data. On frozen teacher-reached states, a 4B student trained with CoRD reaches in 16 steps a gap closure that on-policy distillation does not reach in 80, and substitution rules intervening at least as often in other directions fall short of CoRD.