ReTaCo: Residual-Target Control for On-Policy Distillation

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On-policy distillation (OPD) trains a student on its own generated prefixes with token-level teacher feedback, but transmitting or storing the teacher's full-vocabulary distribution at every token is costly. Entropy-aware OPD (EOPD) adds forward supervision to reverse KL to help the student recover plausible tokens it underestimates, using only the teacher's top-$k$ probabilities to limit cost. Because EOPD renormalizes these probabilities, its target assigns no mass to the omitted vocabulary. We prove that the resulting loss keeps pushing the student's top-$k$ mass toward one even after the student matches the teacher's relative probabilities within the top-$k$ set, so the teacher itself is not a stationary point whenever the omitted tokens have positive teacher probability. We propose ReTaCo (Residual-Target Control), which keeps the top-$k$ tokens individually and groups the remaining tokens into one residual symbol, and pairs this forward target with a single-sample estimator whose expectation equals the full-vocabulary reverse KL. With teacher top-$k$ mass $m$, the residual target is $(1-β)(1-m)$ for $β\in[0,1]$: $β=0$ preserves the teacher's mass, and larger $β$ moves more mass onto the top-$k$ tokens without changing their relative probabilities. At a fixed prefix, we prove that the population objective has a unique optimum whose top-$k$ mass lies between $m$ and $m+β(1-m)$ and increases monotonically with $β$; at $β=0$, underestimated top-$k$ tokens still receive non-vanishing recovery gradients. Numerical optimization confirms these predictions, and across three teacher-student pairs, ReTaCo outperforms EOPD on most mathematics and code benchmarks.