# The Teacher Is a Direction, Not a Destination: Extrapolating RL-Induced Representation Residuals in On-Policy Distillation

arXiv: https://arxiv.org/abs/2609.36484 · HTML: https://mlsanity.com/paper/2609.36484/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in On-policy and self-distillation for LLMs, one of 12 directions in Knowledge distillation.
- Builds on 45 of its 50 references that are in the map.
- First posted on arXiv 29 Sep 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, 30 of 45)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [FitNets: Hints for Thin Deep Nets](https://mlsanity.com/paper/1412.6550/) (2014)
- [MiniLM: Deep Self-Attention Distillation for Task-Agnostic Compression of Pre-Trained Transformers](https://mlsanity.com/paper/2002.10957/) (2020)
- [Representation Engineering: A Top-Down Approach to AI Transparency](https://mlsanity.com/paper/2310.01405/) (2023)
- [TinyBERT: Distilling BERT for Natural Language Understanding](https://mlsanity.com/paper/1909.10351/) (2019)
- [Editing Models with Task Arithmetic](https://mlsanity.com/paper/2212.04089/) (2022)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [Paying More Attention to Attention: Improving the Performance of Convolutional Neural Networks via Attention Transfer](https://mlsanity.com/paper/1612.03928/) (2016)
- [Refusal in Language Models Is Mediated by a Single Direction](https://mlsanity.com/paper/2406.11717/) (2024)
- [On-Policy Distillation of Language Models: Learning from Self-Generated Mistakes](https://mlsanity.com/paper/2306.13649/) (2023)
- [Steering Language Models With Activation Engineering](https://mlsanity.com/paper/2308.10248/) (2023)
- [Sequence-Level Knowledge Distillation](https://mlsanity.com/paper/1606.07947/) (2016)
- [Contrastive Representation Distillation](https://mlsanity.com/paper/1910.10699/) (2019)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [Phi-4-Mini Technical Report: Compact yet Powerful Multimodal Language Models via Mixture-of-LoRAs](https://mlsanity.com/paper/2503.01743/) (2025)
- [Patient Knowledge Distillation for BERT Model Compression](https://mlsanity.com/paper/1908.09355/) (2019)
- [Process Reinforcement through Implicit Rewards](https://mlsanity.com/paper/2502.01456/) (2025)
- [Self-Distilled Reasoner: On-Policy Self-Distillation for Large Language Models](https://mlsanity.com/paper/2601.18734/) (2026)
- [Reinforcement Learning via Self-Distillation](https://mlsanity.com/paper/2601.20802/) (2026)
- [Rethinking On-Policy Distillation of Large Language Models: Phenomenology, Mechanism, and Recipe](https://mlsanity.com/paper/2604.13016/) (2026)
- [Disentangling Length from Quality in Direct Preference Optimization](https://mlsanity.com/paper/2403.19159/) (2024)
- [MiMo-V2-Flash Technical Report](https://mlsanity.com/paper/2601.02780/) (2026)
- [Learning beyond Teacher: Generalized On-Policy Distillation with Reward Extrapolation](https://mlsanity.com/paper/2602.12125/) (2026)
- [RL's Razor: Why Online Reinforcement Learning Forgets Less](https://mlsanity.com/paper/2509.04259/) (2025)
- [Entropy-Aware On-Policy Distillation of Language Models](https://mlsanity.com/paper/2603.07079/) (2026)

## Built on by (most cited first, 1 of 1)

- [SquidAgent: Parallelize Wisely, Coordinate Efficiently](https://mlsanity.com/paper/2610.08647/) (2026)

## Abstract (arXiv)

On-policy distillation (OPD) trains a student to match the teacher's next-token distributions on the student's own trajectories and has yielded substantial empirical gains. Generalized variants allow the student to surpass the teacher by extrapolating an implicit reward in output space. The language-model head, however, attenuates this change anisotropically: much of the change encoded in the teacher's hidden states reaches the logits at a small fraction of its weight, and the sampled-token log-probability ratios on which output-space extrapolation relies inject noise that the extrapolation amplifies, making training unstable. We observe that reinforcement learning (RL) shifts a model's internal representations relative to its base checkpoint, and that the direction of this shift can be measured at every layer. Motivated by this observation, we propose RIDE (RL-Induced Direction Extrapolation), which extrapolates the RL-induced change directly in representation space: at every layer and token position, RIDE computes the residual between the teacher and its pre-RL checkpoint and regresses the student's hidden states toward targets displaced beyond the teacher along this residual. Conditioned on a sampled trajectory, this regression is equivalent to maximizing a linear directional reward defined by the residual under a quadratic penalty centered at the teacher, which makes explicit how the objective moves the student along the RL-induced direction while limiting its deviation from the teacher. Across four base/RL-teacher pairs spanning different scales, architectures, and pre-training lineages, RIDE approaches or exceeds the RL-trained teacher on every pair and is the only method whose mean does so, and it consistently outperforms output-space extrapolation, which degrades the student whenever the teacher is close to its base. Project page: https://github.com/xixixixixxxx/RIDE.