# Divergence controls entropy in distillation

arXiv: https://arxiv.org/abs/2610.03529 · HTML: https://mlsanity.com/paper/2610.03529/

## 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 48 of its 66 references that are in the map.
- First posted on arXiv 2 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, 30 of 48)

- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [The Pile: An 800GB Dataset of Diverse Text for Language Modeling](https://mlsanity.com/paper/2101.00027/) (2020)
- [High-Dimensional Continuous Control Using Generalized Advantage Estimation](https://mlsanity.com/paper/1506.02438/) (2015)
- [Gemma 3 Technical Report](https://mlsanity.com/paper/2503.19786/) (2025)
- [Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling](https://mlsanity.com/paper/2304.01373/) (2023)
- [Tulu 3: Pushing Frontiers in Open Language Model Post-Training](https://mlsanity.com/paper/2411.15124/) (2024)
- [DeepSeek-V4: Towards Highly Efficient Million-Token Context Intelligence](https://mlsanity.com/paper/2606.19348/) (2026)
- [On-Policy Distillation of Language Models: Learning from Self-Generated Mistakes](https://mlsanity.com/paper/2306.13649/) (2023)
- [Scheduled Sampling for Sequence Prediction with Recurrent Neural Networks](https://mlsanity.com/paper/1506.03099/) (2015)
- [Beyond the 80/20 Rule: High-Entropy Minority Tokens Drive Effective Reinforcement Learning for LLM Reasoning](https://mlsanity.com/paper/2506.01939/) (2025)
- [The Entropy Mechanism of Reinforcement Learning for Reasoning Language Models](https://mlsanity.com/paper/2505.22617/) (2025)
- [Self-Distilled Reasoner: On-Policy Self-Distillation for Large Language Models](https://mlsanity.com/paper/2601.18734/) (2026)
- [Understanding the Effects of RLHF on LLM Generalisation and Diversity](https://mlsanity.com/paper/2310.06452/) (2023)
- [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)
- [Reasoning with Exploration: An Entropy Perspective](https://mlsanity.com/paper/2506.14758/) (2025)
- [Self-Distillation Enables Continual Learning](https://mlsanity.com/paper/2601.19897/) (2026)
- [MiMo-V2-Flash Technical Report](https://mlsanity.com/paper/2601.02780/) (2026)
- [Beyond Chinchilla-Optimal: Accounting for Inference in Language Model Scaling Laws](https://mlsanity.com/paper/2401.00448/) (2023)
- [Compact Language Models via Pruning and Knowledge Distillation](https://mlsanity.com/paper/2407.14679/) (2024)
- [Revisiting On-Policy Distillation: Empirical Failure Modes and Simple Fixes](https://mlsanity.com/paper/2603.25562/) (2026)
- [Dissecting Adam: The Sign, Magnitude and Variance of Stochastic Gradients](https://mlsanity.com/paper/1705.07774/) (2017)
- [Verbalized Sampling: How to Mitigate Mode Collapse and Unlock LLM Diversity](https://mlsanity.com/paper/2510.01171/) (2025)
- [Sticking the Landing: Simple, Lower-Variance Gradient Estimators for Variational Inference](https://mlsanity.com/paper/1703.09194/) (2017)
- [Variational Sequential Monte Carlo](https://mlsanity.com/paper/1705.11140/) (2017)

## Abstract (arXiv)

Distillation has become a core primitive of large language model training, but its properties are not yet well understood. We take an entropic perspective, studying how the entropy of the student depends on the data and the divergence that define the distillation objective. We prove that forward KL inflates the entropy of the student above that of the teacher. Since cross-entropy training is a special case, this yields an identity that we verify quantitatively in pretraining and supervised finetuning. Other divergences come with no such guarantee: reverse KL deflates entropy until the gap between student and teacher gets too large, and interpolating between the two changes entropy smoothly early in training but abruptly at convergence. The lower entropy of on-policy distillation comes from token-level reverse KL, not from on-policy sampling. The divergence therefore acts as an implicit entropy regularizer, whose role is clearest in self-distillation: as conditioning on privileged information deflates entropy, the divergence hyperparameters that work best are those that compensate for it.