Per-token gating of forward/reverse KL losses has become a standard technique for on-policy knowledge distillation (OPD), but existing methods such as EOPD (Jin et al., 2026) and ToDi (Jung et al., 2025) each fix a single gating signal and a single gating direction, and the two have never been compared directly. We introduce a four-coefficient parameterization lambda_t = sigma(a * h_t + b * u(x) + c + d * gap_t) in which direction-aligned proxies of EOPD and ToDi appear as one-dimensional (1D) restrictions, and which adds multi-channel composition and an explicit bias as further degrees of freedom. On TweetEval (Barbieri et al., 2020) emotion and hate, with a Qwen3-32B teacher and a Qwen3-4B student, configurations in the full family reach higher accuracy than the matched-magnitude single-channel (entropy-only / gap-only) 1D restrictions in 33 of 36 comparable cells, and a 26-cell mean-match isolation experiment places dynamic gating ahead of effective-KL-matched static baselines in 19 of 26 cells. Because cells share training data, models, and parameter substructure, we report both counts as exploratory aggregate directional evidence rather than as independent hypothesis tests. Targeted three-seed paired replications of the nine headline comparisons singled out by that sweep -- including a third task, offensive -- are directionally consistent, but individually smaller than the single-seed estimates and not significant at n=3. We therefore present the parameterization primarily as a shared coordinate system for comparing per-token gating designs in short-output classification OPD.
On-policy distillation (OPD) trains a student on its own rollouts with token-level teacher supervision. Recent selective OPD methods exploit the non-uniformity of OPD signals by prioritizing high-entropy or high-disagreement tokens. We revisit this principle and ask: which token-level teacher signals are actually learnable? Using a fixed-context diagnostic that measures same-context teacher-student KL reduction, we show that raw KL disagreement is a coarse proxy for learning value. It conflates learnable disagreement, where the teacher assigns corrective mass to the student's top-K candidates, with incompatible disagreement, where the teacher places mass mostly off the student's current support. We formalize this local compatibility as token teachability and show that it better predicts fixed-context improvement than raw KL alone. Motivated by this finding, we propose Teachability-Aware OPD (TA-OPD), a lightweight token-position selection method that applies OPD loss to high-teachability positions without reward models or verifiers. Across Qwen2.5 and Qwen 3 teacher-student settings, TA-OPD often surpasses full-token OPD with only 5% retained tokens and improves over entropy- and divergence-based baselines. Our results reframe selective OPD as selecting learnable teacher signals rather than merely salient tokens.
On-policy distillation (OPD) provides dense token-level supervision by asking a teacher to score student-generated rollouts. However, when the student drifts into an unrecoverable prefix, the teacher may locally agree with the degraded state, producing low reverse KL but little corrective training signal. We identify this persistent regime as a low-KL agreement trap. Further analyses show that tokens during and after such traps produce less useful supervision signals. We propose KAT (KL Agreement Trap Termination), an online OPD termination rule that detects persistent low-KL agreement with a dynamic training-adaptive threshold. By filtering weak supervision from degenerate agreement, KAT improves avg@k accuracy by 2.66% and pass@k by 3.43% across four mathematical benchmarks, while reducing average rollout length by 59.73%.
Knowledge distillation is central to LLM post-training, yet its design space remains poorly understood, especially alongside reinforcement learning (RL). We show that the prevailing paradigms, off-policy distillation and on-policy distillation (OPD), implicitly couple two orthogonal choices: prefix source and token-level KL direction. This follows from decomposing sequence-level KL over autoregressive response distributions: forward KL pairs teacher prefixes with token-level forward KL, and reverse KL pairs student prefixes with token-level reverse KL. We argue this coupling is not intrinsic: decoupling the two axes yields four valid objectives. We establish gradient-level identities showing forward KL gives SFT-style cross-entropy matching with teacher soft targets, whereas reverse KL gives an RL-style policy-gradient objective with a dense teacher-student log-ratio reward, connecting them to off-policy SFT, DAgger-style on-policy SFT, offline-RL-style distillation, and OPD. We conduct an extensive controlled study on math reasoning, evaluating the four objectives both as standalone methods and as initializations for subsequent RL. The results reveal three tradeoffs: KL direction induces an accuracy-entropy tradeoff, prefix source a quality-compute tradeoff, and training length an accuracy-stability tradeoff. Motivated by these findings, we propose KL mixing and an entropy-gated length curriculum. KL mixing shows long-sequence distillation requires substantial forward-KL weight to prevent entropy collapse and length inflation without sacrificing accuracy. The entropy-gated length curriculum improves Avg@k and Pass@k by 3.6 and up to 5.8 points, and cuts average response length by roughly 3x versus fixed long-horizon training. Our results provide a framework and practical methods for designing reasoning distillation objectives that balance accuracy, diversity, compute, and RL behavior.