Lightning OPD 2.0: Mitigating Style Bias in Cross-Teacher On-Policy Distillation for Large Reasoning Models

On-policy distillation (OPD) provides dense token-level supervision on student-generated trajectories. Although OPD performs strongly when teacher and student belong to the same model family, we find that its effectiveness degrades in cross-family settings even after tokenizer alignment, with substantially stronger external teachers offering little additional improvement. To understand this disconnect, we decompose the cross-family OPD signal into two components: an offset between a low-capability teacher-family reference and the student, and the within-family log-likelihood shift from that reference to the strong teacher. Standard OPD transfers both components together, allowing the offset to dominate the update direction and obscure the changes associated with teacher capability improvements. We propose CompassOPD, which removes this offset and transfers the within-family shift, while a frozen student reference anchors updates to the student's initial policy. Thus, both teacher-side and student-side changes are measured within their respective model families. Experiments across three student families and multiple teacher families show that CompassOPD consistently outperforms standard cross-family OPD, improving average reasoning accuracy by up to 5.50 points. For an MoE teacher, we further construct the reference directly from the teacher checkpoint by reducing expert activation, eliminating the need for a separate reference checkpoint while retaining a 3.43-point gain over OPD.

On-policy distillation (OPD) supervises the student exclusively in the output space by matching next-token distributions. This paradigm suffers from two limitations: (i) a high-variance gradient estimator whose signal-to-noise ratio collapses as the student approaches the teacher, and (ii) an LM-head information bottleneck that discards the teacher's intermediate hidden states. We propose On-Policy Representation Distillation (OPRD), the first method to lift on-policy distillation into the hidden-state space. OPRD aligns student and teacher representations across selected layers on the same on-policy rollouts, providing dense, deterministic, per-layer supervision while bypassing the LM head entirely. Theoretically, OPRD provides a deterministic per-sample gradient, removing the token-level estimation variance that plagues OPD, and exposes structural information that any output-space objective necessarily discards. Empirically, OPRD closes the student-teacher gap on competition mathematics benchmarks (AIME 2024, AIME 2025, and AIMO), where every output-space baseline plateaus below the teacher, while training 1.44x faster and using up to 54% less memory. We further extend OPRD to the cross-architecture setting via OPRD-Bridge. By exploiting the observation that heterogeneous models share a low-rank representational structure, we construct a frozen projector pair that aligns representations across arbitrary depth and width mismatches, shifting the alignment from the output space (which depends on a shared vocabulary) to the representation space. We validate OPRD-Bridge on both cross-architecture (Qwen3-4B -> Qwen3-1.7B-Base) and cross-tokenizer (Phi-4-mini-reasoning -> Qwen3-1.7B-Base) settings, demonstrating successful knowledge transfer even when the vocabulary-based alignment channel is unavailable. Code: https://github.com/ShenzhiYang2000/OPRD.

On-Policy Distillation (OPD) improves large language model reasoning by training a student model on trajectories sampled from its own policy under teacher supervision. Although OPD operates on trajectories, its learning signal remains token-level: it identifies deviations through high-loss tokens and repairs them through local reverse-KL correction. We show that this "trajectory-sampled but token-learned" mechanism cannot reliably bridge student trajectories toward teacher trajectories. About 30% of high-loss tokens fall into the low-divergence regime, indicating that many are surface-form mismatches rather than real reasoning forks. Moreover, even truly divergent tokens are difficult to repair with isolated token-level supervision, since reasoning failures often unfold as short-horizon distributional drift. We propose Trajectory-aware OPD (TOPD), which uses near-future trajectory information to identify real divergent states and distribute guidance across multiple future tokens. Experiments show that suppressing non-divergent high-loss tokens improves standard OPD from 47.8% to 48.2% average accuracy, while TOPD further improves performance to 52.2%, with gains on AIME24 from 60.0% to 63.3% and AIME25 from 46.7% to 53.3%.