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

Abstract

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.