TL;DR: Existing on-policy distillation methods that try to surpass the teacher destabilize training by amplifying noise in output space. A new method, RIDE, instead extrapolates the RL-induced representation change directly in representation space — and beats the teacher on all four tested model pairs. Title: The Teacher Is a Direction, Not a Destination: Extrapolating RL-Induced Representation Residuals in On-Policy Distillation URL: Points 🎯 Treats the teacher as a direction, not a destination — regresses the student toward a target that extrapolates the pre/post-RL hidden-state residual ⚠️ The LM head's anisotropic spectrum attenuates most of that change: the weakest 512 head directions carry 79.8% of hidden-state energy but only 30.0% of output-space energy 📉 Output-space extrapolation's variance grows as (λ-1)²; RIDE's gradient variance is 12.6x lower than ExOPD's 📊 Beats the teacher on all 4 model pairs: 56.38 on R1-Distill-1.5B, 66.07 on Qwen3-4B, outperforming OPRD by 0.97–4.06 points ❌ The output-space baseline ExOPD underperforms both the teacher and OPRD on every pair 🔬 Ablations confirm the direction itself matters: random or reversed directions don't deliver the gain that the true RL-induced direction does It's striking how much the choice of where you measure the difference changes whether "surpassing the teacher" actually works. #Distillation# #ReinforcementLearning#