Hybrid Policy Distillation for LLMs

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.

Reasoning-centric large language models (LLMs) achieve strong performance by generating intermediate reasoning trajectories, but often incur excessive token usage and high inference-time decoding cost. We observe that, when solving the same problems, larger reasoning models can often produce more concise traces, whereas smaller reasoning models tend to generate longer and more redundant trajectories. This is especially problematic in real-world deployment, where memory, latency, and serving-cost constraints often favor smaller models. Our observations suggest that reasoning compression can be transferred from large models to small ones rather than enforced through explicit length constraints. Based on this insight, we propose Mixed-Policy Distillation (MPD), a reasoning compression framework that transfers concise reasoning behavior from a larger-sized teacher to a smaller student by distilling teacher-compressed student trajectories. Unlike on-policy distillation, which aligns the student with teacher distributions over verbose student trajectories, or off-policy distillation, which relies on teacher-generated trajectories and may suffer from distribution mismatch, MPD combines the strengths of both. Given a student-sampled trajectory, the teacher rewrites it into a more concise reasoning trace, and the student is trained via KL-based alignment on the compressed trajectory. This preserves student-policy exploration while injecting teacher-guided compression. Experiments on Qwen3-1.7B show that MPD reduces token usage by up to 27.1% while improving performance across multiple reasoning benchmarks, demonstrating an effective approach to efficient small-model reasoning.

Knowledge distillation (KD) transfers knowledge from a large teacher model to a smaller student. In language modeling, the student is trained either on tokens sampled from the teacher (hard labels) or the teacher's full next-token distribution (soft labels). Despite soft labels appear strictly richer, we find that mixing hard and soft labels consistently yields better results. Crucially, we show that this gain cannot be explained by closer teacher matching during training. Instead, it comes from reduced exposure bias, the mismatch between training and inference distributions. To explain this phenomenon, we introduce the Bridge-Garden Decomposition theory, which categorizes generation steps into two types: Bridges, where the next token must be exact, and Gardens, where it can be flexible. We show that hard-only KD excels in Bridges by avoiding risky deviations, while soft-only KD preserves diversity in Gardens. A hybrid strategy handles both cases and, as a result, reduces exposure bias across the sequence. Guided by this theory, we develop a family of Bridge-Garden hybrid supervision methods that adaptively balance hard and soft labels. Across a primary suite of seven teacher-student pairs (including Qwen, Llama, Gemma, and DeepSeek) and benchmarks in reasoning and coding, our approach outperforms divergence-based and on-policy KD baselines while reducing training cost by 9.7x, enabling efficient model compression. Code is available at https://github.com/ghwang-s/bridge_garden_hybrid_kd_release.