Beyond Absolute Imitation: Anchored Residual Guidance for Privileged On-Policy Distillation

On-policy distillation (OPD) has demonstrated strong empirical gains in enhancing complex reasoning in LLMs by aligning a student model with a teacher's predictive distribution over the student's own trajectories. An emerging variant, Privileged OPD, further strengthens this paradigm by employing a self-teacher model augmented with privileged information, such as oracle traces, to mitigate teacher-student capacity gaps while providing dense, answer-directed supervision. However, current methods treat privileged information as a monolithic imitation target, failing to disentangle locally reachable reasoning steps from future-conditioned oracle signals. Consequently, the student is encouraged to match a hindsight-biased distribution that often falls outside its local predictive support. This reachability mismatch incentivizes the student model to skip valid intermediate reasoning in favor of locally unsupported shortcuts. To resolve this, we introduce Anchored Residual On-Policy Distillation (AR-OPD), a dual-view framework that disentangles privileged supervision. Rather than enforcing strict full-view imitation, AR-OPD establishes a locally compatible anchor using a partially privileged teacher, isolating and injecting oracle foresight as a controlled residual to provide destination-directed guidance. Across diverse reasoning tasks, AR-OPD outperforms full privileged OPD by 2.3 points and SFT by 7.9 points. Crucially, this anchored residual mechanism reduces hindsight leakage by 21.7% and mitigates late-stage drift, yielding up to a 7.2-point advantage on challenging long-horizon trajectories exceeding 768 tokens.