Shijia Kang

Haotian Xu, Xing Wu, Weinong Wang et al.

Can scaling transform reasoning? In this work, we explore the untapped potential of scaling Long Chain-of-Thought (Long-CoT) data to 1000k samples, pioneering the development of a slow-thinking model, RedStar. Through extensive experiments with various LLMs and different sizes, we uncover the ingredients for specialization and scale for Long-CoT training. Surprisingly, even smaller models show significant performance gains with limited data, revealing the sample efficiency of Long-CoT and the critical role of sample difficulty in the learning process. Our findings demonstrate that Long-CoT reasoning can be effectively triggered with just a few thousand examples, while larger models achieve unparalleled improvements. We also introduce reinforcement learning (RL)-scale training as a promising direction for advancing slow-thinking systems. RedStar shines across domains: on the MATH-Hard benchmark, RedStar-code-math boosts performance from 66.2\% to 81.6\%, and on the USA Math Olympiad (AIME), it solves 46.7\% of problems using only 21k mixed-code-math datasets. In multimodal tasks like GeoQA and MathVista-GEO, RedStar-Geo achieves competitive results with minimal Long-CoT data, outperforming other slow-thinking systems like QvQ-Preview. Compared to QwQ, RedStar strikes the perfect balance between reasoning and generalizability. Our work highlights that, with careful tuning, scaling Long-CoT can unlock extraordinary reasoning capabilities-even with limited dataset and set a new standard for slow-thinking models across diverse challenges. Our data and models are released at https://huggingface.co/RedStar-Reasoning.

Haotong Yang, Yi Hu, Shijia Kang et al.

Large language models (LLMs) can solve an increasing number of complex reasoning tasks while making surprising mistakes in basic numerical understanding and processing (such as 9.11 > 9.9). The latter ability is essential for tackling complex arithmetic and mathematical problems and serves as a foundation for most reasoning tasks, but previous work paid little attention to it or only discussed several restricted tasks (like integer addition). In this paper, we comprehensively investigate the numerical understanding and processing ability (NUPA) of LLMs. Firstly, we introduce a benchmark covering four common numerical representations and 17 distinct numerical tasks in four major categories, resulting in 41 meaningful combinations in total. These tasks are derived from primary and secondary education curricula, encompassing nearly all everyday numerical understanding and processing scenarios, and the rules of these tasks are very simple and clear. Through the benchmark, we find that current LLMs fail frequently in many of the tasks. To study the problem, we train small models with existing and potential techniques for enhancing NUPA (such as tokenizers, PEs, and number formats), comprehensively evaluating their effectiveness using our testbed. We also finetune practical-scale LLMs on our proposed NUPA tasks and find that 1) naive finetuning can improve NUPA a lot on many but not all tasks, and 2) surprisingly, techniques designed to enhance NUPA prove ineffective for finetuning pretrained models. We further explore the impact of chain-of-thought techniques on NUPA. Our work provides a more detailed and comprehensive understanding of NUPA in LLMs. Our benchmark and code are released at https://github.com/GraphPKU/number_cookbook.

Haotong Yang, Zitong Wang, Shijia Kang et al.

While Large Language Models (LLMs) have demonstrated strong math reasoning abilities through Reinforcement Learning with *Verifiable Rewards* (RLVR), many advanced mathematical problems are proof-based, with no guaranteed way to determine the authenticity of a proof by simple answer matching. To enable automatic verification, a Reward Model (RM) capable of reliably evaluating full proof processes is required. In this work, we design a *scalable* data-construction pipeline that, with minimal human effort, leverages LLMs to generate a large quantity of high-quality "**question-proof-check**" triplet data. By systematically varying problem sources, generation methods, and model configurations, we create diverse problem-proof pairs spanning multiple difficulty levels, linguistic styles, and error types, subsequently filtered through hierarchical human review for label alignment. Utilizing these data, we train a proof-checking RM, incorporating additional process reward and token weight balance to stabilize the RL process. Our experiments validate the model's scalability and strong performance from multiple perspectives, including reward accuracy, generalization ability and test-time guidance, providing important practical recipes and tools for strengthening LLM mathematical capabilities.

Shijia Kang, Muhan Zhang

Reinforcement learning (RL) has been pivotal in enhancing the reasoning capabilities of large language models (LLMs), but it often suffers from limited exploration and entropy collapse, where models exploit a narrow set of solutions, leading to a loss of sampling diversity and subsequently preventing RL from further improving performance. This issue is exacerbated in parallel sampling methods, where multiple outputs are drawn from the same distribution, potentially causing the model to converge to similar solutions. We propose SESA, a novel SEquential SAmpling framework that mitigates this challenge by generating diverse solution sketches sequentially before expanding them into full reasoning paths. This approach ensures broader exploration by conditioning each new output on previous ones, promoting diversity throughout the process and preventing policy collapse. Our experiments on a synthetic task show that sequential sampling consistently outperforms traditional RL methods in terms of path diversity and recovery from collapse. Further evaluations on real-world tasks demonstrate that SESA improves both the exploration of valid strategies and the overall performance of LLMs. On three agent benchmarks, SESA lifts success rates by $+0.25$, $+0.42$, and $+0.07$ absolute over the base model (up to an additional $211\%$ relative improvement over baseline RL), underscoring its exploration advantage. This work introduces a structured approach to exploration, paving the way for more effective and diverse reasoning in RL-trained LLMs. Our code is released at https://github.com/MuLabPKU/sesa.

Zian Li, Xiyuan Wang, Shijia Kang et al.

Invariant models, one important class of geometric deep learning models, are capable of generating meaningful geometric representations by leveraging informative geometric features in point clouds. These models are characterized by their simplicity, good experimental results and computational efficiency. However, their theoretical expressive power still remains unclear, restricting a deeper understanding of the potential of such models. In this work, we concentrate on characterizing the theoretical expressiveness of a wide range of invariant models under fully-connected conditions. We first rigorously characterize the expressiveness of the most classic invariant model, message-passing neural networks incorporating distance (DisGNN), restricting its unidentifiable cases to be only highly symmetric point clouds. We then prove that GeoNGNN, the geometric counterpart of one of the simplest subgraph graph neural networks, can effectively break these corner cases' symmetry and thus achieve E(3)-completeness. By leveraging GeoNGNN as a theoretical tool, we further prove that: 1) most subgraph GNNs developed in traditional graph learning can be seamlessly extended to geometric scenarios with E(3)-completeness; 2) DimeNet, GemNet and SphereNet, three well-established invariant models, are also all capable of achieving E(3)-completeness. Our theoretical results fill the gap in the expressive power of invariant models, contributing to a rigorous and comprehensive understanding of their capabilities.

Yi Hu, Shijia Kang, Haotong Yang et al.

Length generalization, the ability to solve problems longer than those seen during training, remains a critical challenge for large language models (LLMs). Previous work modifies positional encodings (PEs) and data formats to improve length generalization on specific symbolic tasks such as addition and sorting. However, these approaches are fundamentally limited to special tasks, often degrading general language performance. Furthermore, they are typically evaluated on small transformers trained from scratch on single tasks and can cause performance drop when applied during post-training stage of practical LLMs with general capabilities. Hu et al., (2024) proposed Rule-Following Fine-Tuning (RFFT) to improve length generalization in the post-training stage of LLMs. Despite its compatibility with practical models and strong performance, RFFT is proposed for single tasks too, requiring re-training for each individual task with extensive examples. In this paper, we study length generalization in multi-task settings and propose Meta Rule-Following Fine-Tuning (Meta-RFFT), the first framework enabling robust cross-task length generalization. As our first contribution, we construct a large length generalization dataset containing 86 tasks spanning code execution, number processing, symbolic and logical reasoning tasks, beyond the common addition or multiplication tasks. Secondly, we show that cross-task length generalization is possible with Meta-RFFT. After training on a large number of tasks and instances, the models achieve remarkable length generalization ability on unseen tasks with minimal fine-tuning or one-shot prompting. For example, after fine-tuning on 1 to 5 digit addition, our 32B model achieves 95% accuracy on 30 digit addition, significantly outperforming the state-of-the-art reasoning models (DeepSeek-R1-671B: 72%), despite never seeing this task during RF-pretraining.