# Co-RA

The authors propose Residual Advantage (RA), which treats the teacher–student probability residual as a bounded one-step reward, subtracts the corresponding state value under the student policy to form an advantage, and centers it within each response before adding it to the verifier advantage, so the verifier sets a response's mean label while the teacher only redistributes credit among its steps; with Qwen3-1.7B/4B-Base students and a Qwen3-8B teacher, RA combined with GRPO or REINFORCE++ improves all 24 three-seed comparisons on three mathematical benchmarks, raising macro Avg@8 by 1.7–3.6 points and Pass@8 by 3.9–6.3 points, surpassing teacher-only OPD; Co-RA updates a teacher LoRA with verifier advantages on the same scored student batch and adds a further 1.0–1.5 Avg@8 points.

The authors propose Residual Advantage (RA), which treats the teacher–student probability residual as a bounded one-step reward, subtracts the corresponding state value under the student policy to form an advantage, and centers it within each response before adding it to the verifier advantage, so the verifier sets a response's mean label while the teacher only redistributes credit among its steps; with Qwen3-1.7B/4B-Base students and a Qwen3-8B teacher, RA combined with GRPO or REINFORCE++ improves all 24 three-seed comparisons on three mathematical benchmarks, raising macro Avg@8 by 1.7–3.6 points and Pass@8 by 3.9–6.3 points, surpassing teacher-only OPD; Co-RA updates a teacher LoRA with verifier advantages on the same scored student batch and adds a further 1.0–1.5 Avg@8 points.