Let the Data Decide: Supervision Analysis, Capability Trade-offs, and Adaptive Objective Routing in Continued Pre-Training via Off-Policy Distillation

Off-policy distillation is now central to large language model pre-training, yet how training data, objective parameterization, and model capabilities interact remains poorly characterized. We studies top-$k$-truncated, temperature-scaled off-policy distillation by decomposing this problem into two questions: an \emph{objective-to-capability} analysis of how the training objective shapes token-level supervision and downstream performance, and a \emph{data-to-objective} analysis of how data heterogeneity should inform objective routing. We first show that the language-modeling objective ($L_{\mathrm{LM}}$) and the knowledge-distillation objective ($L_{\mathrm{KD}}$) induce systematically different capability profiles, and trace this divergence to a gradient-level tension between \emph{direct observed-token reinforcement} and \emph{teacher-supported alternative supervision}. To quantify this tension, we introduce diagnostic metrics -- support coverage, observed-token probability mass, and teacher-distribution concentration -- and show via controlled sweeps that the support size $k$ governs a coverage-sharpness trade-off, while distillation temperature controls within-support probability allocation. We then examine adaptive objective routing: a domain-level policy that applies $L_{\mathrm{LM}}$ to math and code and $L_{\mathrm{KD}}$ to general-domain data yields consistent gains over both single-objective baselines, whereas token-level routing based on observed-token probability mass or teacher entropy fails to consistently match the single-objective baseline. These results suggest that effective objective routing depends less on routing granularity than on the quality of the routing signal, reframing continued pre-training via off-policy distillation as a structured, data-conditional supervision-design problem rather than a global hyperparameter choice.