DenseSteer: Steering Small Language Models towards Dense Math Reasoning

Large reasoning models (LRMs) have attracted increasing attention for their ability to solve complex mathematical problems by generating extended reasoning chains. In this work, we focus on two critical yet underexplored aspects of the reasoning process: reasoning transitions capturing the distinct transitions between reasoning steps and answer candidates reflecting the variety of solution paths produced by the model. We collectively define these two aspects as thinking schemata. We observe a correlation between the diversity of thinking schemata and model performance, which motivates us to enhance diversity as a means to further improve reasoning potential. To this end, we propose Diverse Schemata Policy Optimization (DiScO), a framework that first endows the model with schemata awareness, then encourages diversity through reinforcement learning, and further promotes diverse reasoning at inference time. Experiments on multiple mathematical reasoning benchmarks demonstrate that DiScO consistently outperforms standard group relative policy optimization. Beyond accuracy, human-annotated analyses show that DiScO substantially improves the model's ability to recover from erroneous initial attempts. Overall, our work suggests the important role that diversity of the thinking schemata plays and points to scaling along the diversity dimension as a promising research direction.

Inference-time computation has greatly enhanced the performance of large language models (LLMs) on challenging reasoning tasks, but this strategy can incur high inference costs. One solution is to route intermediate chain-of-thought (CoT) states to language models of different sizes; however, existing approaches rely on handcrafted routing strategies that limit performance, or on training large process reward models that may be infeasible in many applications. We formulate stepwise model routing as a constrained decision-making problem, which we solve by training a small control policy using reinforcement learning in conjunction with threshold calibration to tune the performance-efficiency tradeoff. We validate our method on three math benchmarks (GSM8K, MATH500, and OmniMath) on both open and closed models. Our method consistently improves the accuracy-cost tradeoff compared to handcrafted approaches, while achieving a comparable tradeoff to methods that require training large process reward models.

The significant computational demands of large language models have increased interest in distilling reasoning abilities into smaller models via Chain-of-Thought (CoT) distillation. Current CoT distillation methods mainly focus on transferring teacher-generated rationales for complex reasoning to student models. However, they do not adequately explore teachers' dynamic attention toward critical information during reasoning. We find that language models exhibit progressive attention shifts towards key information during reasoning, which implies essential clues for drawing conclusions. Building on this observation and analysis, we introduce a novel CoT distillation framework that transfers the teacher's stepwise attention on key information to the student model. This establishes structured guidance for the student's progressive concentration on key information during reasoning. More importantly, we develop a Mixture of Layers module enabling dynamic alignment that adapts to different layers between the teacher and student. Our method achieves consistent performance improvements across multiple mathematical and commonsense reasoning datasets. To our knowledge, it is the first method to leverage stepwise attention within CoT distillation to improve small model reasoning.