# A Privileged Teacher That Only Scores Cannot Search: Critique-Conditioned On-Policy Self-Distillation for Reasoning

arXiv: https://arxiv.org/abs/2610.13744 · HTML: https://mlsanity.com/paper/2610.13744/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in On-policy and self-distillation for LLMs, one of 12 directions in Knowledge distillation.
- Builds on 26 of its 36 references that are in the map.
- First posted on arXiv 9 Oct 2026.

## Placed in

- Knowledge distillation / [On-policy and self-distillation for LLMs](https://mlsanity.com/distill/on-policy-and-self-distillation-for-llms/)

## Builds on (most cited first, 26 of 26)

- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Large Language Models Cannot Self-Correct Reasoning Yet](https://mlsanity.com/paper/2310.01798/) (2023)
- [Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model?](https://mlsanity.com/paper/2504.13837/) (2025)
- [Solving math word problems with process- and outcome-based feedback](https://mlsanity.com/paper/2211.14275/) (2022)
- [Generative Verifiers: Reward Modeling as Next-Token Prediction](https://mlsanity.com/paper/2408.15240/) (2024)
- [Self-Distilled Reasoner: On-Policy Self-Distillation for Large Language Models](https://mlsanity.com/paper/2601.18734/) (2026)
- [The Lessons of Developing Process Reward Models in Mathematical Reasoning](https://mlsanity.com/paper/2501.07301/) (2025)
- [Rethinking On-Policy Distillation of Large Language Models: Phenomenology, Mechanism, and Recipe](https://mlsanity.com/paper/2604.13016/) (2026)
- [DeepMath-103K: A Large-Scale, Challenging, Decontaminated, and Verifiable Mathematical Dataset for Advancing Reasoning](https://mlsanity.com/paper/2504.11456/) (2025)
- [Logic-RL: Unleashing LLM Reasoning with Rule-Based Reinforcement Learning](https://mlsanity.com/paper/2502.14768/) (2025)
- [When Can LLMs Actually Correct Their Own Mistakes? A Critical Survey of Self-Correction of LLMs](https://mlsanity.com/paper/2406.01297/) (2024)
- [On Memorization of Large Language Models in Logical Reasoning](https://mlsanity.com/paper/2410.23123/) (2024)
- [TIP: Token Importance in On-Policy Distillation](https://mlsanity.com/paper/2604.14084/) (2026)
- [GPT-4 Doesn't Know It's Wrong: An Analysis of Iterative Prompting for Reasoning Problems](https://mlsanity.com/paper/2310.12397/) (2023)
- [Critique Fine-Tuning: Learning to Critique is More Effective than Learning to Imitate](https://mlsanity.com/paper/2501.17703/) (2025)
- [Not All Disagreement Is Learnable: Token Teachability in On-Policy Distillation](https://mlsanity.com/paper/2605.26844/) (2026)
- [RLCSD: Reinforcement Learning with Contrastive On-Policy Self-Distillation](https://mlsanity.com/paper/2606.11709/) (2026)
- [Can Large Language Models Detect Errors in Long Chain-of-Thought Reasoning?](https://mlsanity.com/paper/2502.19361/) (2025)
- [Rethinking On-Policy Self-Distillation for Thinking Models](https://mlsanity.com/paper/2607.05184/) (2026)
- [When Are Teacher Tokens Reliable? Position-Weighted On-Policy Self-Distillation for Reasoning](https://mlsanity.com/paper/2605.21606/) (2026)
- [Learning to Self-Verify Makes Language Models Better Reasoners](https://mlsanity.com/paper/2602.07594/) (2026)
- [Privileged, but Biased: How PI-Conditioned Teachers Break Self-Distillation](https://mlsanity.com/paper/2608.04794/) (2026)
- [Rebellious Student: Reversing Teacher Signals for Reasoning Exploration with Self-Distilled RLVR](https://mlsanity.com/paper/2605.10781/) (2026)
- [Is More Privileged Information Better? From Solution Traces to Problem-Solving Structure in Self-Distilled Reasoning](https://mlsanity.com/paper/2608.01589/) (2026)
- [Finding the Evidence: Discovering Decision-Supporting Tokens for On-Policy Reasoning Distillation](https://mlsanity.com/paper/2606.22830/) (2026)
- [Localizing Credit at the Divergence: Path-Conditioned Self-Distillation for LLM Reasoning](https://mlsanity.com/paper/2606.15576/) (2026)

## Abstract (arXiv)

On-policy self-distillation promises to replace the sparse outcome signal of verifiable-reward RL with dense token-level credit: a teacher conditioned on the reference solution scores the student's own rollout, and the resulting distributional shift is distilled back into the policy. We investigate whether this scoring signal actually contains corrective credit. Using gold error spans in student rollouts, we test the two signals consumed by self-distillation objectives, the full-distribution KL and the token log-ratio, for two properties: whether they localize errors and whether they push probability away from erroneous student tokens. They exhibit neither of these properties nor can they reliably distinguish correct from incorrect rollouts within the same problem. We argue that this failure is structural: locating an error in a reasoning trace is itself a search problem, while a scoring teacher is never explicitly asked to perform that search. The same model, with the same privileged information, localizes errors substantially better when allowed to generate a critique. This forms the motivation for critique-conditioned self-distillation: first generate a critique of the current rollout, then condition the scoring teacher on that critique. The resulting signal localizes errors, is corrective within error spans, and shifts probability toward tokens that initiate backtracking such as ``Wait'', ``But'', and ``Actually''. We further show that the critique is useful both at test time, where resuming a wrong rollout with the critique in context converts math problems the model cannot solve by sampling, and in cold-start GRPO, where it provides an advantage signal on prompts for which verifiable rewards alone provide none.