Language Model Distillation

Knowledge distillation can transfer reasoning from stronger teachers to frozen students through reusable prompts, but avoiding weight updates does not eliminate supervision. Without ground-truth answers, teacher solutions are unverified, and agreement with the teacher can reward shared mistakes. We introduce Knowledge-to-Prompt (K2P) for label-free knowledge distillation to prompts. K2P synthesizes reusable instructions from teacher solutions, refines them using paired teacher and student responses, and guides search and selection with answer agreement. It retains candidates that adaptive search may undervalue and selects on reserved questions. Deployment uses only the frozen student and selected prompt. Our theory separates generation and selection gaps and gives conditions under which agreement-guided construction yields accuracy guarantees despite imperfect teacher references. Across reasoning tasks and students, K2P outperforms label-free alternatives overall and remains competitive with supervised prompt optimization. Ablations and archive diagnostics assess the contributions of teacher solutions and refinement, while revealing the limits of agreement-guided selection.

We study training LLM judges from natural language feedback, especially for subjective tasks where the verdict depends strongly on which evaluation criteria the judge invokes and how it weighs them. The dominant approach, outcome-supervised RL (e.g., GRPO), credits every token in the rollout with a single scalar determined only by the accuracy of the final verdict, providing no separate credit at the criterion-choice tokens and ignoring the rich language feedback (e.g., preference rationales) that naturally accompanies preference labels. Self-Distillation (SD) is one natural way to use this language feedback: the same model, conditioned on this feedback, acts as a teacher providing dense, position-level supervision. However, not all positions carry equally useful signal. Using the per-position entropy shift between teacher and student, we identify two regimes: context sharpening, where the teacher concentrates probability on a particular feedback-aligned criterion expression, and context spreading, where the teacher distributes probability across multiple feedback-aligned alternatives. We interpret these patterns as follows: sharpening encourages memorization of a particular criterion expression, whereas spreading promotes semantic understanding by preserving these alternatives. Motivated by this asymmetry, we introduce position masking based on the entropy shift that retains the lower tail of the entropy-shift distribution. Experiments show that masking higher-entropy-shift positions improves out-of-distribution generalization over naive SD. The resulting self-distilled judges outperform judges trained with outcome-supervised RL by 2-9 percentage points on the evaluated subjective subcategories, while remaining competitive on objective ones.

On-policy distillation (OPD) has recently emerged as a promising post-training paradigm in which the student learns from trajectories generated by its own policy under dense teacher supervision. However, OPD introduces a fundamental asymmetry: although the sampled trajectories are on-policy for the student, they are off-policy for the teacher. The teacher is typically optimized to continue from prefixes generated by its own policy, but during OPD it must instead supervise prefixes generated by the student. Empirically, we find that its continuation performance degrades as these prefixes grow longer. To address this issue, we propose Student-COnditioned Updates of the Teacher (SCOUT), a co-training framework that adapts the teacher to student-generated prefixes. Alongside standard OPD updates, SCOUT periodically optimizes the teacher's conditional ability using reinforcement learning with verifiable rewards, where the teacher generates continuations from student prefixes and learns from outcome rewards. Controlled experiments show that SCOUT improves the teacher's ability to continue from student-generated prefixes, supporting the intended mechanism of student-conditioned teacher adaptation. Across multiple teacher--student configurations, model scales, and reasoning domains, SCOUT also consistently improves the effectiveness of on-policy distillation.

On-policy self-distillation uses a model as its own teacher to provide dense supervision for reasoning, often through reference-solution conditioning. Providing privileged information does not by itself ensure effective token-level supervision throughout long responses. We introduce Activation-Conditioned Self-Distillation (ACSD), which extracts a steering vector by contrasting activations of self-generated trajectories that reach verified correct answers within a generation budget with those of all remaining trajectories. A frozen copy of the base model applies this vector at each prediction position, and the student learns from its next-token distributions on student-generated prefixes. Outcome verification is used for direction construction and calibration; distillation requires neither problem-specific reference text nor teacher parameter updates. The distilled student is used alone at inference. On each of five models, ACSD achieves the highest mean accuracy over four mathematical benchmarks among the evaluated methods. On DeepSeek-R1-0528-Qwen3-8B, mean mathematical accuracy reaches 71.9% and LiveCodeBench v6 pass@12 reaches 70.9%, compared with 69.0% and 66.3% for the reference-conditioned OPSD baseline. Contrasts among correct trajectories also support distillation, and extracted directions can be reused across mathematical training datasets. On fixed student trajectories, ACSD maintains more stable late-position logit-update magnitudes than OPSD.

On-policy distillation (OPD) trains a student on its own generated responses using dense, token-level supervision from a stronger teacher. Vanilla OPD treats all teacher signals equally, assuming that the teacher's supervision is equally important for every token. However, teacher signals at different tokens may have very different effects on the student's performance: some correct important reasoning errors, while others have little effect on the final answer. Motivated by this observation, we introduce Dr. OPD (OPD Done Right), which defines the optimal weighted OPD to maximize the student's performance. We formulate Dr. OPD as a bilevel optimization problem in which the student learns from weighted teacher supervision, while the weights are selected to maximize the expected reward of the resulting student. To solve Dr. OPD, we develop an efficient iterative solver that updates the token weights and student policy alternatively. At each round, it updates weights in closed form and then takes one gradient step on the resulting weighted OPD objective. Under regularity conditions, we show that this weighted update achieves a higher expected reward than a vanilla OPD update. Empirically, across strong-to-weak and same-size distillation on math and code, Dr. OPD consistently outperforms all evaluated baselines. In particular, in the strong-to-weak distillation setting, Dr. OPD improves average math performance by $9.7$ points over vanilla OPD, and enables the smaller student to surpass its larger teacher.

On-policy distillation (OPD) trains a student on its own reasoning trajectories using feedback from a stronger teacher. Teacher interventions can improve these trajectories, but also change the distribution on which the student learns. Our controlled studies show that rollout quality alone is an incomplete criterion for allocating teacher guidance. Deeper intervention yields diminishing gains in rollout accuracy while increasing off-policy load. In a training probe with a restricted rollout horizon, peak student accuracy and performance retention favor different intervention strengths. The preferred intervention depth and placement also vary across benchmarks. These findings motivate MAESTRO, which uses local policy disagreement to jointly adapt when the teacher takes over and how long it generates. Its {policy disagreement score} combines teacher-weighted candidate coverage with local distribution similarity and is aggregated within reasoning paragraphs. Across eight mathematical reasoning benchmarks, MAESTRO achieves the highest macro-average accuracy among the compared methods for both 0.6B and 1.7B Qwen3 students, with the 1.7B student leading on every benchmark. MAESTRO also reduces average training response length by 67.3% relative to standard OPD. The code is available at https://github.com/yhao-wang/MAESTRO.

On-policy distillation (OPD) trains students using teacher feedback on their own sampled responses, yet how prompt choice shapes transfer across teacher-student pairs remains poorly understood. We systematically study prompt quantity, source, and selection across RL- and SFT-continuation pairs and cross-model settings. We find that OPD can be highly prompt-efficient: a few prompts can approach large-pool performance, with four DAPO prompts matching the observed mathematics score of 3,840 DeepMath prompts. However, prompt utility is relational rather than intrinsic: changing only the teacher can reverse the relative effectiveness of mathematics and code prompts. To characterize these transfer differences, we analyze parameter and functional changes across prompt supports and model pairs. Functional alignment with the teacher varies across supports and target tasks; in continuation pairs, teacher-aligned prediction changes can coexist with weak parameter alignment. Continued OPD on effective supports can restore performance after unfavorable transfer. Finally, targeted selection does not consistently outperform uniform random sampling, and filtering out a source that performs poorly alone yields no consistent gain across three paired support draws. Overall, our results distinguish prompt efficiency from prompt interchangeability and show that effective data choice depends on the teacher-student pair and target capability, with random sampling providing a competitive baseline in the studied settings.

Off-policy and on-policy distillation have traditionally been formulated as separate paradigms, each favoring a different property of distillation trajectories. Teacher-generated (off-policy) traces are typically high-quality but lie far from the student's distribution, whereas student-generated (on-policy) rollouts are more learnable but often contain erroneous reasoning. We view these paradigms as the endpoints of a policy continuum and posit that a more effective rollout policy may lie in between. We introduce \textbf{Interpolated Policy Distillation (IPD)}, which defines the next-token distribution at every decoding step as an explicit linear interpolation between the student and teacher distributions. The interpolation operates at the distribution level, token by token, and its coefficient provides direct control over the balance between trajectory quality and student learnability. Naively sampling from this policy would require sequentially querying the teacher at every token and is thus expensive. To make IPD practical, we accelerate it with a new speculative-decoding rule while exactly preserving the interpolated next-token distribution.At the trajectory level, the resulting rollouts naturally interleave student- and teacher-generated segments. Unlike recent heuristic segment-interleaving methods, however, this interleaving is induced by an exactly realized token-level interpolated policy rather than by hand-designed switching rules. Across text-only and multimodal reasoning benchmarks, IPD consistently outperforms both endpoint policies (SFT and OPD), their conventional two-stage combination (SFT-then-OPD), and recent heuristic segment-interleaving methods, demonstrating that token-level policy interpolation better balances trajectory quality and student learnability.

On-policy self-distillation (OPSD) improves large language models by letting a self-teacher with privileged information provide dense token-level supervision on the model's own trajectories. Yet existing methods typically construct the self-teacher from the current, initial, or slowly averaged policy state, leaving the quality of supervision constrained by the teacher's ability to exploit privileged information. We ask whether the model's own optimization progress can instead be recycled into a stronger self-teacher. In this paper, we introduce Bootstrapped On-Policy Self-Distillation (B-OPSD), which temporarily trains the policy ahead to obtain a future teacher, restores the student to the original policy state, and then uses the future teacher to supervise the restarted student. The future teacher improves supervision in two complementary ways, it can generate more reliable privileged trajectories and, conditioned on them, provide more informative token-level targets along the restarted student's on-policy trajectories. Experiments on mathematical reasoning with Qwen3-4B and Qwen3-8B show consistent improvements over standard OPSD in both settings, including gains from 27.50 to 41.30 and from 48.80 to 64.44 in the rollout-privileged setting. Our findings point to a broader principle for self-improving models that future learning progress can be distilled backward, preserving acquired knowledge while bootstrapping beyond the optimization state that produced it.

Post-training is central to mathematical reasoning in modern large language models (LLMs), but endpoint pass@1 alone underidentifies what has changed. Gains may reflect newly reachable solutions, cheaper sampling of latent solutions, surface robustness, or memorisation. We compare three post-training paths under a common diagnostic readout: our sufficiently trained off-policy distillation trajectories, released Qwen3 off-policy-plus-on-policy distillation endpoints, and a released DeepSeek-Math endpoint trained with Group Relative Policy Optimisation (GRPO). Our probe uses cross-surface pass@K over verbatim prompts, paraphrases, numerical isomorphisms, and translations, plus consistency, distribution-shape, and verified supervised-fine-tuning (SFT) membership analyses. We find two regimes. On easier AMC problems, large-K ceilings are near saturation, so post-training mainly compresses sample cost. On harder AIME problems, post-training expands the large-K ceiling over the base model: sufficient off-policy distillation already raises this ceiling, Qwen3 released endpoints raise it further, and DeepSeek-Math GRPO does not dominate sufficient off-policy distillation at large K. English-dominant distillation improves non-English reasoning but preserves language-tier gaps. A controlled-overfit audit finds limited sensitivity in current SFT-membership probes. Compression is one regime of post-training, not a universal explanation.

Explicit intermediate reasoning gives large language models (LLMs) a stronger problem-solving mode. We study learning from this think-mode advantage via on-policy distillation (OPD). OPD preserves student-generated trajectories and provides dense token-level teacher targets at student-visited prefixes. Privileged reasoning is used during distillation rather than student inference. Uniform ThinkOPD, a natural think-enabled OPD baseline, conditions a fixed teacher on one shared think trace and uniformly distills every sibling student response. Although its prefixes are on-policy, the trace need not follow a route compatible with every complete response: the same privileged trace can induce different teacher-student discrepancies even when responses reach the same outcome. We summarize this interaction with trace-response divergence (TRD) and introduce ThinkOPD, which routes supervision at the response level by combining group-relative reward gain with a TRD-based compatibility proxy. Final response weights are normalized within each rollout group. Across mathematical reasoning and code generation, ThinkOPD outperforms Uniform ThinkOPD in both same-model settings and both cross-model teacher-student pairs, and it exceeds representative rationale and self-distillation baselines in a controlled comparison. Controlled interventions show that outcome benefit and the TRD-based proxy provide complementary routing signals in this setting. Think-enabled OPD provides a controlled setting for studying how teacher advantage becomes transferable along student responses.

Self-Distillation Fine-Tuning (SDFT) enables a language model to act as its own teacher: by conditioning on a demonstration, the model produces an implicit reward via pointwise mutual information, which guides on-policy learning without external supervision. However, SDFT operates at training time: it requires gradient updates and access to expert demonstrations, making it inapplicable at inference. We propose test-time self-distillation, a decoding-time method that extracts a steering signal from the self-distillation framework without any parameter updates, reward models, or training data. Our key insight is that counterfactual contexts, i.e. fixed textual templates that hypothetically prime the model for excellent versus poor reasoning, can substitute for the demonstration. The log-odds ratio of a candidate answer under these two counterfactual conditions defines a new reward signal. We derive the optimal KL-regularized policy under this reward, which takes the form of a Gibbs reweighting of the base distribution. Crucially, this reweighting is global: it cannot be decomposed into independent per-token operations without ignoring future trajectory quality. We therefore approximate the target distribution via beam search. Experiments on mathematical reasoning (MATH500), code generation (HumanEval), and graduate-level science QA (GPQA) across multiple model scales show that test-time self-distillation improves over standard sampling, low temperature, beam search and power sampling baselines on average, demonstrating that the self-distillation principle can be operationalized at inference time.

Self-distillation turns knowledge distillation into a closed learning loop and offers a path toward recursive self-improvement. Without an external teacher, however, the model must determine both what information can improve its supervision and which induced changes should be learned. Existing methods typically improve teacher-generated data or select training examples in isolation, leaving the information transferred between these stages unmeasured. We introduce InFlow, a retrieval-guided on-policy self-distillation framework that models this process as potential-to-realized information flow. InFlow first retrieves potentially informative sources using certainty-calibrated hidden-state trajectories, then measures their realized effect through the Jensen--Shannon divergence between the teacher's initial and retrieval-conditioned answer beliefs. Examples with larger belief shifts are selected for on-policy distillation. Our analysis formalizes the information optimized by retrieval and selection and relates the answer-level shift to the teacher--student distillation gap. Across four open-weight language models and three knowledge domains, InFlow achieves the strongest cross-model average among the compared selection methods, with ablations supporting both stages of the framework. Our code is available at https://github.com/1240148048/INFLOW.

On-policy distillation (OPD) reduces train-test state mismatch by training a student on its own generated trajectories, but weak students may visit teacher-misaligned prefixes where supervision is less representative. We introduce SAKI (Supervision Allocation with KL-constrained Interpolation), which combines a KL-constrained teacher-guided rollout with maximal coupling and reuses realized accept/correction events to route token-level supervision. Accepted positions retain sampled-token reverse-KL supervision, while correction positions receive direct supervision on the teacher's highest-probability token. Under maximal coupling, the correction probability is exactly TV(p_t, q_t), so the same trust-region radius controls rollout deviation and upper-bounds intervention and specialized-supervision frequency. We further implement an engine-resident speculative verifier that preserves the exact-q trajectory distribution and coupling semantics while improving matched-workload rollout throughput by 4.22x. Across seven mathematical reasoning benchmarks, SAKI improves the matched teacher-guided baseline in Mean@8 and Pass@8 for both 1.7B and 0.6B students. Placement controls and fixed-prefix analysis further support correction-triggered routing as a conflict-adaptive supervision signal.

On-policy distillation (OPD) trains a student model on its self-generated trajectories with dense token-level teacher feedback. However, naive OPD may suffer from teacher unanchoring, where the student's reasoning trajectory drifts far from the teacher, causing the teacher to be queried on states it would hardly visit and thus provide unreliable supervision. We propose Interactive-Policy Distillation (IPD), which applies adaptive teacher intervention to the student rollout. Under a bidirectional propose-and-verify state machine, the student and teacher alternately exchange their roles as proposer and verifier, and collaboratively generate mixed-source trajectories. Then different supervisions are applied according to the source of each token. This bidirectional propose-and-verify mechanism and the source-split loss make IPD not only a more performant distillation method, but also a unified bridge between on-policy and off-policy paradigms. To make the interleaved dual-model rollouts more efficient, we also design a dedicated fused inference engine that co-hosts both models in one serving instance with separate KV caches and instantiates the state machine model to distribute, collect, and process requests. On math reasoning tasks and across multiple teacher-student model pairs, student models trained with IPD not only outperform those trained with OPD, but also demonstrate higher data efficiency. Specifically, when distilling Qwen3-30B-A3B into Qwen3-1.7B-Base, IPD brings a +3.28 mean@8 and a +3.28 best@8 benchmark-averaged accuracy improvement compared with OPD. Besides, IPD only consumes about 1/4 of the training examples and steps to outperform OPD trained on the whole training dataset for one epoch. We also investigate the impact of different loss variants and takeover / handback configurations, and demonstrate the robustness of IPD on different training data.

We present OLIVE (OnLine InterVEntion). At each iteration, the evolving student policy generates a new prefix, the teacher continues it autoregressively, and the student is updated using cross-entropy computed on the teacher-generated tokens. Each design choice targets a corresponding limitation of existing distillation methods: (1) sequential covariate shift in offline supervised fine-tuning (SFT) on fixed teacher trajectories, (2) fragmented supervision under prefix failure in token-level on-policy distillation (OPD), and (3) the need for access to teacher token probabilities in distribution-matching distillation. OLIVE achieves higher reasoning performance than OPD (with a top-16 KL approximation) at comparable GPU-hour cost. Our asynchronous implementation further reduces OLIVE's total training time by 23.8%. We evaluate OLIVE on both hard reasoning tasks and agentic tasks which reflects modern post-training scenarios, and it consistently outperforms existing distillation methods under the same training budget. By regenerating prefixes from the evolving student, OLIVE continues improving after offline distillation plateaus while better preserving the general capabilities and plasticity of the student. Using only text from GPT-5.4-mini, continuously training with OLIVE outperforms offline SFT from the same teacher by 13% on ScienceWorld. These results support OLIVE as an effective and efficient approach to online language-model distillation.

Distillation attacks copy the reasoning capabilities of closed-source large language models, allowing bad actors to replicate state-of-the-art performance at low cost. Attackers systematically collect a large volume of frontier model reasoning traces and then train (i.e., "distill") their own models on these traces. Existing defenses against distillation attacks are typically evaluated immediately after distillation, implicitly assuming attackers do not train their models any further. In this paper, we argue that a more realistic threat model includes further training with reinforcement learning after distillation. A misspecified threat model can give a false sense of security -- some defenses that seem effective after distillation can be broken after subsequent reinforcement learning. Practically, reinforcement learning lowers the bar for a distillation attack to be effective. We show that simple attacks can steal reasoning capabilities from existing closed-source language models using data easily obtainable from current APIs, yielding reasoning improvements equivalent to more sophisticated attacks that extract the full hidden traces. Results indicate that any distillation defense that leaks sufficient information to reconstruct approximate reasoning traces is likely ineffective. We conclude by discussing broader implications and batch-level distillation defenses which could be more effective.

On-policy distillation (OPD) trains a student language model with dense feedback from a stronger teacher on student-generated trajectories. Yet standard OPD weights token-level distillation terms uniformly, implicitly treating local teacher preference as a proxy for correction utility. A decision's task value, however, depends on how the student completes the subsequent reasoning. This mismatch can cause imitation to suppress viable student strategies or reinforce paths the student cannot reliably execute. Verified trajectory outcomes provide complementary evidence about continuation quality, but do not directly identify the utility of individual decisions. We introduce Reward-Aligned Reweighting for On-Policy Distillation (R$^{2}$-OPD), which uses outcome agreement and the magnitude of teacher--student disagreement to continuously reallocate teacher supervision. It gives reward-aligned corrections greater relative influence while retaining dense feedback, moving beyond uniform imitation and hard filtering. Our analysis formalizes the mismatch between local teacher preference and student continuation value and establishes sufficient conditions for reallocation to improve first-order task progress over uniform OPD. Across seven mathematical reasoning benchmarks, R$^{2}$-OPD achieves the highest average accuracy among the compared training methods in both cross-size and same-size distillation. It outperforms standard OPD on all seven benchmarks, with average gains of 3.5 and 2.4 percentage points for 1.7B and 4B students, respectively. An extension to code generation yields an average gain of 1.6 percentage points over standard OPD. These results highlight outcome-guided supervision allocation as an effective way to translate dense teacher feedback into stronger student performance across model scales and task domains.

We study on-policy distillation (OPD) through the lens of reinforcement learning, establishing a connection between the reverse-KL objective in OPD and KL-regularized policy optimization. Building on this connection, we introduce Least-Square Policy Distillation (LSPD), an RL-inspired framework that brings optimistic exploration and off-policy data reuse from value-based RL into policy distillation. LSPD preserves policy diversity through exploration while improving rollout efficiency by repeatedly learning from previously collected trajectories. Our theoretical analysis connects LSPD to optimistic value-based learning and shows that its idealized formulation achieves a sharp $\tilde{\mathcal O}(\log K)$ regret bound under online exploration. Empirically, LSPD consistently outperforms existing distillation baselines across six mathematical reasoning benchmarks and diverse teacher-student settings, with average gains of +1.59 points in Avg@16. Remarkably, through Pass@k evaluations up to k=64, we found that LSPD better preserves policy diversity by achieving stronger performance as k grows. Its fully off-policy variant achieves comparable performance to vanilla OPD using only the first 25% of rollout batches. Together, these results provide an RL perspective on OPD that offers both a principled interpretation and a practical route toward more effective and rollout-efficient language model distillation.

Verbal feedback can identify errors and prescribe corrections, providing rich supervision for language-model post-training even when reliable programmatic verifiers are unavailable. Such feedback, often generated by a capable model, can be used to condition the teacher in on-policy distillation, which trains the student to match the teacher's predictions on student-generated rollouts. However, this approach can transfer teacher preferences that the feedback did not motivate, while leaving much of the feedback's guidance unused. We find that both problems come from the standard on-policy distillation objective, specifically the divergence it minimizes and the distribution it uses as its target. Our proposed method addresses both limitations. First, to isolate the information conveyed by the feedback from the teacher's inherent preferences, we treat verbal feedback as evidence for or against the hypothesis that a particular token comes next at a given prefix. We then adopt a probabilistic confirmation framework which uniquely determines an ordering over the vocabulary based on the teacher's predictions before and after it receives feedback. Using a confirmation score consistent with this ordering, we construct a target distribution within a trust region of the student. Second, to learn from guidance that student rollouts can leave unused, we derive a simple shared-rollout estimator of a symmetric divergence between the student and target distributions over rollouts, reusing student and feedback-conditioned teacher rollouts in both directions through importance weighting. Empirical evaluations show that our method outperforms the common on-policy distillation recipe and a recent contrastive variant on knowledge-based and agentic benchmarks.

Reinforcement learning can turn one language model into several specialists, each excellent at a single skill such as mathematics, coding or following instructions, but users need one model with all of these skills. Multi-teacher on-policy distillation (MOPD) merges them by letting the specialists teach one student: the student answers each prompt, and the specialist for that prompt's domain gives feedback on every token. This routing decides which specialist teaches, but not how strongly its feedback moves the shared student. In Qwen3.5 models at three sizes, we find that MOPD's student does not beat one taught by the best single specialist and gains little of the mathematics specialist's advantage. The feedback is unbalanced: instruction-following feedback is several times more spread out than mathematics feedback and dominates the student's updates. We propose Domain-Normalized MOPD (DN-MOPD), which keeps the routing and rescales each domain's feedback by its measured spread. On six public benchmarks, DN-MOPD improves the average score over MOPD at every size, across three random seeds and under two answer-length limits, and recovers most of the lost mathematics gain. Controls with fixed domain weights show that the gain comes mainly from turning down instruction-following feedback rather than turning up mathematics alone, and that fixed weights close to those DN-MOPD measures perform comparably. Combining specialists therefore requires deciding not only which one teaches, but also how strongly its feedback counts.

On-policy distillation (OPD) trains a student on its own trajectories with dense teacher supervision. Recent work on OPD for multi-turn autonomous agents often treats large teacher-student token-level distributional gaps as promising intervention points, linking larger gaps to a greater need for correction. Yet, our empirical analysis reveals a supervision-benefit mismatch: large gaps can be benign, while small gaps can be outcome-critical. Teacher-student gaps capture differences at the current turn, whereas the benefit of teacher guidance depends on how the current student interacts with the environment afterward. The student may still succeed despite choosing an action that differs from the teacher's, while a teacher-preferred action may lead to a state from which the student cannot complete the task. Local gaps alone are therefore not enough to determine whether teacher guidance benefits the current student. Effective supervision should instead emphasize guidance that the current student can translate into better final task outcomes. Accordingly, we propose Outcome-Guided On-Policy Distillation (OG-OPD), which applies trajectory-relative weighting to teacher supervision and calibrates these weights using final task outcomes from paired student continuations. This calibration selectively strengthens supervision on the student's original trajectories at turns where teacher guidance benefits the current student. Across ALFWorld, ScienceWorld, and WebShop, OG-OPD consistently outperforms baselines under diverse settings. It improves task success rates by 3.6-17.7 percentage points over vanilla OPD and by up to 7.0 percentage points over the strongest baseline.

On-policy learning has been argued to reduce catastrophic forgetting, produce sparser parameter updates, and improve generalisation. However, existing comparisons between supervised fine-tuning and reinforcement learning vary many factors simultaneously, making the contribution of rollout policy difficult to isolate. We study the effect of rollout policy in a controlled strong-to-weak distillation setting, by independently varying rollout policy, token-level KL direction, and learning rate across the Llama3 and Qwen2.5 model families and reasoning tasks spanning scientific, medical, and arithmetic domains. Our analysis reveals a nuanced picture of distillation dynamics in which rollout policy does not necessarily play a central role. Instead, token-level KL direction more clearly shapes task performance and output coverage, while learning rate governs forgetting and update sparsity. Analysis of KL gradients and experiments along a continuous student-teacher rollout-policy spectrum explain this pattern: forward KL is remarkably robust to rollout policy, with its performance stable and strong despite changes to the rollout policy, whereas reverse KL is substantially more sensitive and favours student-generated rollouts. On-policy data nevertheless improves generalisation to harder variants of the Countdown arithmetic task under both KL directions, although this advantage does not reliably persist after subsequent RLVR. Our broader conclusions remain robust to removing gradient clipping, using sampled KL estimators, and training on tasks requiring longer reasoning chains. Overall, our results challenge the view that on-policy rollouts are inherently preferable and show that their value depends critically on the objective, evaluation setting, and optimisation hyperparameters.

Multi-teacher on-policy distillation (MOPD) combines independently developed domain teachers into a single student by distilling their predictions on student-generated samples. We study a setting where teachers share a reference model but undergo different post-training procedures, and find that MOPD can struggle to recover some teacher capabilities. Because distillation occurs on student-generated prefixes, the student initialization can strongly affect subsequent recovery. However, initial benchmark performance is not a reliable predictor of a good MOPD initialization. For example, merge initialization can start below SFT warm-up yet finish higher after MOPD. We further find that effective merging depends on both the relative teacher contributions and the overall merge scale, with some strong configurations lying outside the simplex of convex parameter averaging. Thus, selecting a good merge initialization requires evaluating not only its immediate performance but also the learning it enables under MOPD, making one-shot coefficient search difficult. We propose Iterative Merging for MOPD (IM-MOPD), which starts from a uniform merge and progressively adds task-vector increments for under-recovered domains during distillation. In a 5-domain setting, IM-MOPD achieves higher average normalized recovery than MOPD with either uniform merge initialization or SFT warm-up, showing that effective teacher contributions can be determined progressively during training.

Multi-teacher on-policy distillation (MOPD) has emerged as a popular post-training paradigm for integrating specialized capabilities in frontier language models. Existing OPD research has primarily focused on optimizing single-task distillation through objective design, distillation scope, and teacher signal construction, whereas MOPD must aggregate multiple capabilities in shared parameters and address the resulting capability seesaw, in which improving one domain suppresses capabilities acquired from another. Inspired by the distinctive update geometry of OPD, we find that parameter updates from different tasks rapidly concentrate in their respective low-dimensional subspaces during MOPD, providing a direct geometric basis for identifying and controlling cross-task interference. We therefore propose PMOPD (Projection-based Multi-Teacher On-Policy Distillation), which constructs subspace memories from the cumulative parameter displacements of different tasks and projects both gradients and optimizer updates to remove components that interfere with protected task directions. We further develop a lightweight conflict probe to characterize task interactions and guide task ordering, together with a cycling strategy that balances subspace estimation and timely task revisitation. Experiments on representative Code, Reason, and Math tasks show that PMOPD improves every evaluated capability over MOPD, raising the average score across the three tasks by 2.54 points on Qwen2.5-7B and 2.09 points on Llama-3.1-8B. These consistent gains establish geometry-aware optimization as an effective and transferable approach to balanced multi-teacher distillation.

On-policy distillation (OPD) is becoming an important component of large language model (LLM) post-training for transferring the reasoning capability of a strong teacher LLM to a weaker student LLM. OPD trains the student by minimizing the reverse KL divergence between the teacher and the student via rollouts generated by the student's policy. However, estimating the gradient of the reverse KL divergence in OPD remains a challenge. Using only the sampled token from the student-generated rollout is computationally cheap but provides limited distributional supervision, which will degrade accuracy. In addition, using the full vocabulary provides complete distributional supervision but is computationally expensive. Therefore, recent works propose Top-$k$ OPD (TK-OPD) that use selected top-$k$ tokens, which provides richer distributional supervision than sampled-token estimation at substantially lower computational cost than full-vocabulary estimation. Unfortunately, using only the selected top-$k$ tokens induces bias, leading to accuracy degradation, as the probability mass outside the selected top-$k$ tokens is discarded. To address the bias of TK-OPD, we propose Tail-Corrected Top-$k$ On-Policy Distillation (TT-OPD). It preserves the advantages of TK-OPD, including rich distributional supervision and low computational cost, while providing an unbiased estimator of the gradient of the reverse KL divergence. The key insight of TT-OPD is to use not only the selected top-$k$ tokens, but also the sampled token from the student-generated rollout, thereby recovering the discarded probability mass in expectation, avoiding the bias. Experimental results demonstrate that TT-OPD significantly outperforms other tested OPD variants.

On-policy distillation (OPD) uses teacher correction on student-generated responses. Full-vocabulary correction can provide important corrections even for tokens that the student assigns low probability, but backpropagating through all token logits becomes memory-intensive for long sequences. Existing memory-saving approaches estimate corrections from sampled tokens or restrict supervision to the student's TopK tokens, introducing sampling noise or changing the full-vocabulary correction. We introduce \textbf{SparseOPD}, which uses full-vocabulary teacher correction to determine which corrections matter before selecting the token logits to differentiate. SparseOPD first constructs the full-vocabulary correction without retaining its backward graph, then selects tokens by correction magnitude rather than student probability. Signed residual compensation preserves the total promoting and suppressing correction mass, while correction-aware budget allocation distributes the sparse support across positions. Finally, the update backpropagates only through the selected token logits. Across six task--scale settings spanning mathematics, chemistry QA, and multimodal reasoning, SparseOPD outperforms Sampled Token and TopK in task-average accuracy and matches or exceeds Full Vocabulary. Gradient cosine similarity reaches 99% on 4B mathematics, while 8K full-parameter profiling shows 70.5% lower backward memory.

Pretrained autoregressive Transformers represent a large sunk investment in compute. Existing conversion methods reuse that investment by changing either the architecture (attention to recurrence) or the objective (next-token prediction to denoising), never both. We convert Qwen3 teachers at 1.7B and 8B into attention-free, bidirectional, gated-delta-rule diffusion students in three stages, so that each capability can be traced to the stage that kept or lost it. Language modeling transfers only partially and in-distribution; in-context retrieval does not transfer. On a multi-query recall probe where the teachers score 0.34-0.58, both converted students score 0.000, and diffusion pretraining alone does not restore retrieval. A retrieval curriculum in the final stage, which gradually lengthens the gap between a key-value table and the queries that address it, restores it only stochastically: on a fixed schedule, one seed in three learns to retrieve. Advancing the gap only while a running accuracy estimate stays above a threshold works for all three of those seeds, holds on real text, and carries unchanged to 8B, where two of three seeds succeed. The third had not learned within its fixed 16k-step budget: retrieval switches on abruptly at a seed-dependent step (6.5k and 11k in the other two), so a fixed budget can cut a late run off. One boundary survives every intervention: every model that learns retrieval scores 0.000 on tokens that never appeared in a retrieval episode, and an arm that resamples the key and value tokens every batch shows this is a coverage limit, not memorization of particular bindings. Separately, we convert a 7B code model into a 3:1 recurrent-attention block-diffusion hybrid over 85k steps and report two negative training results.

On-policy distillation (OPD) bridges teacher supervision and student behavior, but different teacher-student tokenizers introduce misalignment in both input tokenization (#1) and output logits (#2). Existing approaches address the former by matching same-text spans or converting tokens to bytes, often losing fine-grained token information or disrupting the native-token paradigm, while for the latter, strategies such as ranking, padding, or key-token selection retain only shared logit dimensions, resulting in much distribution loss. In this paper, we propose Dual-Vocabulary Language Model (DVLM), which replaces the teacher's LM head with a new student-vocabulary projection head and obtains full-dimensional student logits (for #2). To support student tokens (for #1), it takes a Parallel-Tokenized Sequence (PTS) as input, which concatenates the original teacher-tokenized sequence and a re-tokenized sequence formed by independently converting each student token into a teacher-token group. To avoid inference inconsistency with the original teacher tokens, the Hybrid-Prefix Attention (HPA) further restricts re-tokenized groups to their corresponding teacher prefix and uses its last state as the aggregation of the original student-token representation for projection into the student vocabulary space. Similarly, via the combined use of PTS and HPA, the DVLM teacher can provide distribution-aligned supervision with the student's input-tokenization and output-logit during OPD. Experimental results demonstrate that our DVLM teacher has a similar converged loss as the original teacher model and enables student models to improve performance across six reasoning tasks.

Since the advent of knowledge distillation, KL divergence has been the standard loss in distillation. Recently, on-policy distillation (OPD) has emerged as an efficient post-training paradigm for LLMs. As a distillation method, OPD naturally inherits KL divergence as its standard loss. However, in this work, we find that KL divergence may not be necessary for OPD. We show that simply preserving the update direction is sufficient for effective OPD. As long as the update direction is toward the teacher, OPD works. More precisely, it is not the direction of every token, but the direction of a small subset of tokens where the teacher and student disagree strongly. We first show that simply assigning a reward of (+1) to tokens where the teacher probability is higher than the student probability and (-1) where it is lower, which merely encourages updates toward the teacher, reproduces almost the same training mode as OPD with reverse KL. We further show that only the direction of a small subset of tokens with large teacher-student disagreement is critical, and training works as long as their update direction is toward the teacher, even if other tokens are pulled away from the teacher. And as an application of these findings, we introduce Consensus Multi-Teacher On-Policy Distillation (C-MOPD) to improve Multi-Teacher On-Policy Distillation (MOPD). Unlike MOPD, which routes each sample to a single teacher and may cause capability conflicts across domains, C-MOPD lets every sample be supervised by all teachers. Experiments show that C-MOPD consistently outperforms MOPD on both math and code benchmarks. Our code is available at https://github.com/LeapLabTHU/KL-Free-OPD.