# A Shared Learning Rate Is Not a Neutral Control in Selective On-Policy Distillation

arXiv: https://arxiv.org/abs/2609.22109 · HTML: https://mlsanity.com/paper/2609.22109/

## 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 17 of its 20 references that are in the map.
- First posted on arXiv 17 Aug 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, 17 of 17)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Scaling Laws for Reward Model Overoptimization](https://mlsanity.com/paper/2210.10760/) (2022)
- [Rethinking On-Policy Distillation of Large Language Models: Phenomenology, Mechanism, and Recipe](https://mlsanity.com/paper/2604.13016/) (2026)
- [Self-Consuming Generative Models Go MAD](https://mlsanity.com/paper/2307.01850/) (2023)
- [TIP: Token Importance in On-Policy Distillation](https://mlsanity.com/paper/2604.14084/) (2026)
- [MOPD: Multi-Teacher On-Policy Distillation for Capability Integration in LLM Post-Training](https://mlsanity.com/paper/2606.30406/) (2026)
- [Understanding and Mitigating Numerical Sources of Nondeterminism in LLM Inference](https://mlsanity.com/paper/2506.09501/) (2025)
- [Not All Disagreement Is Learnable: Token Teachability in On-Policy Distillation](https://mlsanity.com/paper/2605.26844/) (2026)
- [SelecTKD: Selective Token-Weighted Knowledge Distillation for LLMs](https://mlsanity.com/paper/2510.24021/) (2025)
- [DOPD: Dual On-policy Distillation](https://mlsanity.com/paper/2606.30626/) (2026)
- [Cornerstones or Stumbling Blocks? Deciphering the Rock Tokens in On-Policy Distillation](https://mlsanity.com/paper/2605.09253/) (2026)
- [SWITCH: Studying with Teacher for Knowledge Distillation of Large Language Models](https://mlsanity.com/paper/2410.19503/) (2024)
- [Rethinking Selective Knowledge Distillation](https://mlsanity.com/paper/2602.01395/) (2026)
- [Outcome-Confounded Local Supervision in On-Policy Distillation](https://mlsanity.com/paper/2607.23731/) (2026)

## Abstract (arXiv)

Selective on-policy distillation trains a student only at the token positions a selector scores highest, and the literature compares selectors under a single shared learning rate--a control chosen to be neutral. We show it is not. Under LoRA on GSM8K (Qwen2.5-1.5B student, 7B teacher), across an 8x learning-rate grid, dense supervision is statistically flat (swing 1.8 pp, p=0.26) while every selective arm moves with the rate: 5.4 pp for a random 5% subset, 6.7 pp for a total-variation selector, up to 17.7 pp for a teachability selector. Consequently the dense-versus-selective verdict reads 10.1 pp at lr=1e-4 but 5.1 pp at 5e-5--a 2.0x difference decided by a parameter the protocol treats as scenery--and two of six pairwise significance calls between selectors flip between adjacent rates without any rank inversion. We call this selector-rate entanglement and trace it to selection itself rather than step size: AdamW update magnitudes track the rate to within 2.2% despite 15.5x gradient-norm differences across arms. A preregistered frozen-scoring ablation (selection scored by the initial student; criterion, budget, and on-policy rollouts unchanged; 12 seeds per cell) shows live scoring adds 3.79+/-1.69 pp of rate sensitivity (p=0.035) while the frozen arm remains significantly entangled (p=0.015): the feedback loop aggravates the phenomenon rather than causing it. Under full fine-tuning at the rates this literature actually uses (1e-6 to 1e-5) the pattern grows: dense itself swings 19.8 pp, the selective arm 49.5 pp, and the verdict ranges from a non-significant +3.6 pp at the published operating point to +34 pp (p=0.005) one notch hotter. On MATH-500 the rate dependence does not reproduce under LoRA, scoping that result, while the ~10 pp cost of selective training does. We prescribe reporting the arm x rate matrix, not a shared-rate column, as a precondition for selector comparisons.