On-policy distillation is a practical post-training recipe for large language models, supplying dense teacher supervision on the student's own trajectories. In privileged-context self-distillation, teacher and student are the same model conditioned on the same prefix, but the teacher also sees a hint or the full solution trace. This makes supervision abundant but harder to trust: the teacher can be confident about continuations its privileged view makes obvious but the student cannot yet justify. The distillation pull is strongest where teacher and student disagree most, and over many updates it accumulates into drift that degrades out-of-distribution (OOD) reasoning. We introduce GeoSD, a geometric self-distillation objective that treats this drift as movement in the student's predictive behavior and counters it in two complementary ways. A Hellinger loss scales each teacher preference by the overlap the student already shares with it, attenuating the pull on tokens the student cannot yet support. Since these pulls still compound over training, a proximal term penalizes how far the student's predictions drift from a recent checkpoint, measured as a Fisher-Rao distance. Both are distances in the same geometry of next-token distributions, and a natural-gradient update takes its steps in that geometry rather than in parameter space. Across mathematical reasoning benchmarks and three model families, GeoSD preserves the in-distribution gains of self-distillation while improving average OOD accuracy by 5.7-8.6 points over the base model, with gains holding across model scales from 1.7B to 32B. Analyzing why standard matching fails out of distribution, we find it wins agreement with the teacher by draining mass from alternatives at high-entropy states, resulting in confident agreement on wrong answers, whereas GeoSD keeps those alternatives in reach.
We study on-policy self-distillation (OPSD), where a language model improves its reasoning ability by distilling privileged teacher distributions along its own on-policy trajectories. Despite its promise, OPSD can suffer from training instability due to a pattern mismatch between teacher and student responses. Self-reflected teacher responses may introduce reflection-induced biases and response templates that miscalibrate token-level supervision, ultimately harming the student's reasoning ability. To mitigate this issue, we propose OGLS-SD, an outcome-guided logit-steering framework that leverages verifiable outcome rewards to calibrate privileged teacher logits. Specifically, OGLS-SD contrasts teacher logits induced by successful and failed on-policy trajectories, constructing an outcome-discriminative steering direction for token-level guidance. Experiments on mathematical reasoning benchmarks show that OGLS-SD stabilizes self-distillation and improves performance over standard OPSD and other variants.
On-policy self-distillation, where a student is pulled toward a copy of itself conditioned on privileged context (e.g., a verified solution or feedback), offers a promising direction for advancing reasoning capability without a stronger external teacher. Yet in math reasoning the gains are inconsistent, even when the same approach succeeds elsewhere. A pointwise mutual information analysis traces the failure to the privileged context itself: it inflates the teacher's confidence on tokens already implied by the solution (structural connectives, verifiable claims) and deflates it on deliberation tokens ("Wait", "Let", "Maybe") that drive multi-step search. We propose Anti-Self-Distillation (AntiSD), which ascends a divergence between student and teacher rather than descending it: this reverses the per-token sign and yields a naturally bounded advantage in one step. An entropy-triggered gate disables the term once the teacher entropy collapses, completing a drop-in replacement for default self-distillation. Across five models from 4B to 30B parameters on math reasoning benchmarks, AntiSD reaches the GRPO baseline's accuracy in 2 to 10x fewer training steps and improves final accuracy by up to 11.5 points. AntiSD opens a path to scalable self-improvement, where a language model bootstraps its own reasoning through its training signal.
On-policy self-distillation (SD) improves LLM reasoning by using teacher-side privileged information (PI) to turn sparse verifier outcomes into dense token-level supervision. Existing methods usually assume trusted PI, such as reference answers or successful traces. We ask whether PI can instead come from an experience-derived skill bank, where retrieved skills are compact and reusable but may also be irrelevant or misleading. We propose Skill-Conditioned Gated Self-Distillation (SGSD), which formulates skill-based SD as teacher hypothesis validation rather than unconditional imitation. SGSD retrieves skill-mistake pairs, constructs a multi-teacher pool, and lets all skill-conditioned teachers score the same plain-prompt student rollout. The verifier validates each teacher's polarity: supporting a success or suppressing a failure gives positive supervision, while the opposite stance is reversed. A robust gated objective then distills informative teacher-student disagreements while suppressing uncertain or extreme signals. Experiments on multiple mathematical reasoning benchmarks show that SGSD consistently improves over GRPO and remains competitive with answer-conditioned OPSD under a weaker PI assumption. For example, on Qwen3-1.7B, SGSD outperforms GRPO by 6.2% and OPSD by 1.7% on average on AIME24, AIME25, and HMMT25.