Abstract
On-policy self-distillation (OPSD) improves reasoning by training a model on its own trajectories with dense token-level supervision from a privileged copy conditioned on a reference solution. Existing methods typically use a single teacher distribution conditioned on the complete reference. We identify a limitation of this formulation: a reference is only one realization of the reasoning process. Although it provides evidence about the answer, it also encodes an ordering, phrasing, and solution path. Full-reference conditioning can therefore produce correct targets that depend excessively on information unavailable to the student at inference. We show that replacing one valid reference with another, while holding the problem and student state fixed, can substantially alter the teacher distribution, a phenomenon we call reference-induced target instability. To address this instability, we introduce Monte Carlo Privileged Distillation (MCPD), which models privileged context as a distribution over partial reference views and distills the marginal teacher distribution. MCPD evaluates the teacher under multiple views and averages the resulting next-token distributions. We show that this probability-space average is the exact target induced by minimizing the expected teacher-to-student KL divergence over privileged views. Across mathematical and scientific reasoning benchmarks, we show that MCPD produces more reference-consistent supervision and consistently improves over full-reference OPSD.
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URL
https://arxiv.org/abs/2610.30880
