Test-time scaling has emerged as a critical avenue for enhancing the reasoning capabilities of Large Language Models (LLMs). Though the straight-forward ''best-of-$N$'' (BoN) strategy has already demonstrated significant improvements in performance, it lacks principled guidance on the choice of $N$, budget allocation, and multi-stage decision-making, thereby leaving substantial room for optimization. While many works have explored such optimization, rigorous theoretical guarantees remain limited. In this work, we propose new methodologies to predict and improve scaling properties via tail-guided search. By estimating the tail distribution of rewards, our method predicts the scaling law of LLMs without the need for exhaustive evaluations. Leveraging this prediction tool, we introduce Scaling-Law Guided (SLG) Search, a new test-time algorithm that dynamically allocates compute to identify and exploit intermediate states with the highest predicted potential. We theoretically prove that SLG achieves vanishing regret compared to perfect-information oracles, and achieves expected rewards that would otherwise require a polynomially larger compute budget required when using BoN. Empirically, we validate our framework across different LLMs and reward models, confirming that tail-guided allocation consistently achieves higher reward yields than Best-of-$N$ under identical compute budgets. Our code is available at https://github.com/PotatoJnny/Scaling-Law-Guided-search.
A Tale of Tails: Model Collapse as a Change of Scaling Laws
Meta FAIR；Center for Data Science, New York University（纽约大学数据科学中心）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Courant Institute, New York University（纽约大学柯朗研究所）
Effective Frontiers: A Unification of Neural Scaling Laws
Department of Mathematics and Statistics, Xi'an Jiaotong University（西安交通大学数学与统计学系）；Department of Artificial Intelligence, Gaoling School of Artificial Intelligence, Renmin University of China（中国人民大学人工智能学院）；University of Chinese Academy of Sciences（中国科学院大学）；Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences（中国科学院深圳先进技术研究院）
Spectral Reach: Understanding Neural Scaling as Progress into the Spectral Tail
Institute for Computational Physics, University of Stuttgart, Stuttgart, Germany（斯图加特大学计算物理研究所）；Department of Applied Mathematics（应用数学系）；Department of Physics, University of Cambridge, Cambridge, UK（剑桥大学物理系）
