SPEAR: Distilling Domain-Adaptive Reasoning Skeletons via Sequential Symbolic Alignment in Reinforcement Learning
Authors: Zhuochun Li, Yuelyu Ji, Yiming Zeng, Daqing He
Organizations: University of Pittsburgh, Pittsburgh, USA · University of Connecticut, Storrs, USA
Abstract
Reinforcement learning-based knowledge distillation has the potential to transfer complex reasoning from teacher to student models, yet it currently faces a critical dilemma: researchers must choose between sparse outcome-based rewards, which provide insufficient logical guidance, or expensive neural Process Reward Models (PRMs) for dense signals. We resolve this by introducing SPEAR (Symbolic Process Evaluation and Alignment Reward), a training-free and plug-and-play process reward method for sequence-level on-policy distillation. SPEAR projects natural-language reasoning traces into domain-adaptive symbolic milestones, providing an efficient proxy for process-level reasoning alignment. By utilizing the longest common subsequence (LCS) to align student explorations with teacher milestones, SPEAR provides a dense, order-aware reward signal that enforces logical consistency without the need for an external neural verifier. Our experiments across math, science, and commonsense reasoning tasks demonstrate that SPEAR effectively bridges the reasoning gap between student and teacher models via sequence-level distillation with efficient dense process rewards. Our code and data are available at: https://github.com/zhuochunli/SPEAR.
On-policy distillation is an alternative post-training method in reinforcement learning that alleviates the constraints imposed by reward models by providing token-level supervision from a teacher model. Although on-policy distillation has been studied and applied across various settings, its fundamental design remains underexplored. In this paper, we introduce a new distillation reward, termed the delta signal, instead of directly imitating the teacher's output distribution. The delta signal is defined as the difference between the teacher model and its base model prior to instruction tuning for reasoning capability. It therefore captures the changes induced by reasoning tuning and provides a more direct signal for transferring reasoning capabilities. Using extensive empirical evidence, we show that the delta signal substantially improves on-policy distillation and refer to the new distillation method as On-Policy Delta Distillation (OPD2). Experiments across mathematics, science, and code-reasoning benchmarks demonstrate that OPD2 consistently outperforms conventional on-policy distillation, enabling reasoning LLMs to achieve strong performance with only a short post-training period. Code will be available at https://github.com/naver-ai/opd2
On-policy distillation transfers reasoning ability through dense token-level supervision, yet the nature of the transferable signal remains unclear. We discover that reasoning chains contain two types of knowledge that require different discovery mechanisms: decisions (where to branch), which surface through student uncertainty, and evidence (intermediate steps that justify decisions), which hides in positions where the student is confident yet wrong. Current methods capture only decisions; the substantive knowledge in evidence tokens remains untransferred. We propose DEAR(Decision-Evidence Aware Reasoning Distillation), which first identifies decisions via student entropy, then discovers their supporting evidence through hidden-state cosine similarity to decision anchors, boosted by teacher-student divergence to prioritize the largest knowledge gaps. Across three student-teacher configurations on math and code benchmarks, DEAR consistently outperforms standard OPD, with up to +2.5pp on competition math and +5.7pp on code generation.
Distilling reasoning capabilities from Large Reasoning Models (LRMs) into smaller models is typically constrained by the limitations of rejection sampling. Standard methods treat the teacher as a static filter, discarding complex "corner-case" problems where the teacher fails to explore valid solutions independently, thereby creating an artificial "Teacher Ceiling" for the student. In this work, we propose Hindsight Entropy-Assisted Learning (HEAL), an RL-free framework designed to bridge this reasoning gap. Drawing on the educational theory of the Zone of Proximal Development (ZPD), HEAL synergizes three core modules: (1) Guided Entropy-Assisted Repair (GEAR), an active intervention mechanism that detects critical reasoning breakpoints via entropy dynamics and injects targeted hindsight hints to repair broken trajectories; (2) Perplexity-Uncertainty Ratio Estimator (PURE), a ratio-based filtering heuristic that reduces high-anomaly shortcut-like rationales; and (3) Progressive Answer-guided Curriculum Evolution (PACE), a three-stage distillation strategy that organizes training from foundational alignment to hard-case adaptation. Extensive experiments on multiple benchmarks demonstrate that HEAL significantly outperforms traditional SFT distillation and other baselines.