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Abstract: On-policy distillation (OPD) improves a student language model by training on its own generated trajectories with supervision from a stronger teacher. However, when conditioned on student-generated prefixes, the teacher can favor invalid continuations that violate the question's premises, causing OPD to encourage erroneous reasoning steps. As a remedy, we introduce Condition-Contrastive Extrapolated On-Policy Distillation (CCE-OPD), which makes teacher supervision sensitive to question conditions. CCE-OPD incorporates (1) a fixed-prefix comparison in which the same frozen teacher evaluates the complete question and a counterpart with a selected condition masked, and (2) a bounded log-probability contrast that reweights the teacher distribution into an auxiliary distillation target. On 200 diagnostic reasoning states, condition contrast reduces the rate of preferring invalid continuations from 27.0% to 16.0%. Extensive experiments across six mathematical benchmarks show that CCE-OPD improves avg@8 over OPD by an average of 1.60 and 1.95 percentage points when distilling Qwen3-8B into 0.6B and 1.7B students, respectively. It also achieves the strongest results on four transfer benchmarks.

Abstract: On-policy distillation (OPD) transfers knowledge from a teacher model to a student along student-generated trajectories, typically by minimizing the reverse Kullback-Leibler (RKL) divergence. However, RKL can provide weak corrective signals for teacher-supported tokens to which the student assigns very low probability. This raises two questions: which positions are most consequential for the behavioral gap between the student and teacher, and what additional supervision is needed at these positions? Through cross-sampling, we find that divergence-based routing more effectively pinpoints positions contributing to the teacher-student performance gap than entropy-based routing. Further analysis reveals teacher-mode under-coverage at high Jensen-Shannon (JS) divergence positions, where standard OPD increases the large teacher-student probability gaps. Motivated by these findings, we propose Divergence-Routed On-Policy Distillation (DrOPD). DrOPD retains RKL supervision at all positions and selectively adds forward Kullback-Leibler (FKL) divergence supervision at positions with high JS divergence. Experiments across six math benchmarks show that DrOPD improves over standard OPD and entropy-based routing. DrOPD also reduces severe under-coverage cases by 37.9 percent relative to OPD, supporting targeted FKL as an effective complement to OPD.

Abstract: Latent reasoning lets a large language model (LLM) think in a continuous space and verbalize only the answer. We argue that an effective latent thought must meet five requirements: it should be useful, helping produce the correct answer rather than merely changing it, diverse, so that resampling yields different reasoning trajectories, explainable, so that a decoded chain of thought (CoT) reflects reasoning the answer actually follows, refinable with more inference compute, and efficient, costing less than an explicit CoT at comparable accuracy. Current methods rarely meet these requirements: they learn shortcuts from the question, distill the explicit CoT into their weights, or imitate it one token at a time. We focus on flow matching in a learned latent space, the family we argue is best placed to meet them, and identify the training choices that make it work. The result is Flow-based Latent Reasoning (FLaRe), a simple recipe covering what the latent space encodes and how to shape it, where to train the flow, how to read out the answer, and a final stage of training on the model's own verified thoughts. A probe for each requirement shows that FLaRe improves on prior latent methods in all five. It also compares favorably with them on arithmetic benchmarks, while reaching 97% of the accuracy of explicit CoT at a quarter of its latency.