# Reinforcement Learning-based Semi-supervised Knowledge Distillation with LLM-as-a-Judge

arXiv: https://arxiv.org/abs/2604.02621 · HTML: https://mlsanity.com/paper/2604.02621/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 28 of its 40 references that are in the map.
- First posted on arXiv 3 Apr 2026.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)
- 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, 28 of 28)

- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Judging LLM-as-a-Judge with MT-Bench and Chatbot Arena](https://mlsanity.com/paper/2306.05685/) (2023)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [High-Dimensional Continuous Control Using Generalized Advantage Estimation](https://mlsanity.com/paper/1506.02438/) (2015)
- [Solving math word problems with process- and outcome-based feedback](https://mlsanity.com/paper/2211.14275/) (2022)
- [CodeGen: An Open Large Language Model for Code with Multi-Turn Program Synthesis](https://mlsanity.com/paper/2203.13474/) (2022)
- [Are NLP Models really able to Solve Simple Math Word Problems?](https://mlsanity.com/paper/2103.07191/) (2021)
- [Large Language Models Can Be Easily Distracted by Irrelevant Context](https://mlsanity.com/paper/2302.00093/) (2023)
- [Sequence-Level Knowledge Distillation](https://mlsanity.com/paper/1606.07947/) (2016)
- [Rubrics as Rewards: Reinforcement Learning Beyond Verifiable Domains](https://mlsanity.com/paper/2507.17746/) (2025)
- [PandaLM: An Automatic Evaluation Benchmark for LLM Instruction Tuning Optimization](https://mlsanity.com/paper/2306.05087/) (2023)
- [LLaDA 1.5: Variance-Reduced Preference Optimization for Large Language Diffusion Models](https://mlsanity.com/paper/2505.19223/) (2025)
- [Principled Reinforcement Learning with Human Feedback from Pairwise or K-wise Comparisons](https://mlsanity.com/paper/2301.11270/) (2023)
- [ReFT: Reasoning with Reinforced Fine-Tuning](https://mlsanity.com/paper/2401.08967/) (2024)
- [CODI: Compressing Chain-of-Thought into Continuous Space via Self-Distillation](https://mlsanity.com/paper/2502.21074/) (2025)
- [Learning by Playing - Solving Sparse Reward Tasks from Scratch](https://mlsanity.com/paper/1802.10567/) (2018)
- [Fine-Tuning of Continuous-Time Diffusion Models as Entropy-Regularized Control](https://mlsanity.com/paper/2402.15194/) (2024)
- [Beyond Binary Rewards: Training LMs to Reason About Their Uncertainty](https://mlsanity.com/paper/2507.16806/) (2025)
- [Keeping Your Distance: Solving Sparse Reward Tasks Using Self-Balancing Shaped Rewards](https://mlsanity.com/paper/1911.01417/) (2019)
- [Fine-Tuning Discrete Diffusion Models with Policy Gradient Methods](https://mlsanity.com/paper/2502.01384/) (2025)
- [ENGINE: Energy-Based Inference Networks for Non-Autoregressive Machine Translation](https://mlsanity.com/paper/2005.00850/) (2020)
- [Learning Approximate Inference Networks for Structured Prediction](https://mlsanity.com/paper/1803.03376/) (2018)
- [Unlocking Anticipatory Text Generation: A Constrained Approach for Large Language Models Decoding](https://mlsanity.com/paper/2312.06149/) (2023)

## Built on by (most cited first, 3 of 3)

- [Rubric-based On-policy Distillation](https://mlsanity.com/paper/2605.07396/) (2026)
- [Distill Where the Student Goes: Teacher-Regularized RL for English-Evidence Cross-Lingual RAG](https://mlsanity.com/paper/2607.02966/) (2026)
- [Radioactive Feedback: Tracing Models Trained by Label-Only LLM Supervision](https://mlsanity.com/paper/2610.15685/) (2026)

## Abstract (arXiv)

Reinforcement Learning (RL) substantially improves the reasoning capabilities of language models, but most existing RL fine-tuning approaches rely entirely on ground-truth verifiable rewards and thus labeled datasets with verifiable answers. To overcome this, we propose a RL framework for reasoning distillation that leverages continuous, LLM-based rewards. Our method employs an efficient mechanism that computes a continuous CoT reward (CCR) directly from a single-token logit of a judge LLM, evaluating the student model's reasoning trajectory. This formulation provides an effective and scalable online training signal that can be applied to large volumes of unlabeled data. We demonstrate that, when paired with a strong judge, simply using CCR achieves performance comparable to that of ground-truth or pseudo-label verifiable rewards, and even surpasses them as the amount of unlabeled data increases. Furthermore, we find that combining them in a semi-supervised setup is highly synergistic: verifiable rewards help stabilize the CCR, while CCR improves the generalizability of verifiable rewards to related tasks. We also provide a comprehensive empirical comparison of various reward sources across multiple model architectures and dataset sizes. Our results show that this semi-supervised approach consistently enhances mathematical reasoning, yielding an absolute improvement of 5-10% on multiple reasoning tasks.