Plan and Budget: Effective and Efficient Test-Time Scaling on Large
  Language Model Reasoning
Authors: name: Junhong Lin, name: Xinyue Zeng, name: Jie Zhu, name: Song Wang, name: Julian Shun, name: Jun Wu, name: Dawei Zhou

Large Language Models (LLMs) have achieved remarkable success in complex
reasoning tasks, but their inference remains computationally inefficient. We
observe a common failure mode in many prevalent LLMs, overthinking, where
models generate verbose and tangential reasoning traces even for simple
queries. Recent works have tried to mitigate this by enforcing fixed token
budgets, however, this can lead to underthinking, especially on harder
problems. Through empirical analysis, we identify that this inefficiency often
stems from unclear problem-solving strategies. To formalize this, we develop a
theoretical model, BBAM (Bayesian Budget Allocation Model), which models
reasoning as a sequence of sub-questions with varying uncertainty, and
introduce the E^3 metric to capture the trade-off between correctness and
computation efficiency. Building on theoretical results from BBAM, we propose
Plan-and-Budget, a model-agnostic, test-time framework that decomposes complex
queries into sub-questions and allocates token budgets based on estimated
complexity using adaptive scheduling. Plan-and-Budget improves reasoning
efficiency across a range of tasks and models, achieving up to +70% accuracy
gains, -39% token reduction, and +187.5% improvement in E^3. Notably, it
elevates a smaller model (DS-Qwen-32B) to match the efficiency of a larger
model (DS-LLaMA-70B)-demonstrating Plan-and-Budget's ability to close
performance gaps without retraining. Our code is available at
anonymous.4open.science/r/P-and-B-6513/.