BOSS: Token-Budgeted Online Sampling and Selection for LLM Reasoning

Sampling multiple responses improves language model reasoning, but uniform compute allocation is inefficient because easy questions are over-sampled while hard questions remain under-explored. We propose Uncertainty-Aware Budget Allocation (UAB), a concave integer optimization framework that reallocates a fixed sampling budget using uncertainty estimated from the initial samples themselves. In Phase-1, every question receives a small fixed number of generations. Their answer disagreement, measured by vote entropy, provides a difficulty signal while these generations contribute to the final vote. In Phase-2, the remaining budget is allocated by a marginal-greedy algorithm that optimally solves a concave coverage-maximization surrogate, concentrating samples on questions whose initial answers disagree. Across five open-weight models (1.5B--27B parameters) and five reasoning benchmarks of varying difficulty, UAB improves average accuracy by $+2.3\%$ over uniform allocation, and achieves gains of up to $+5.5\%$ on individual benchmarks, with the largest gains in low-resource settings. Moreover, UAB meets the target budget exactly and requires no auxiliary model or additional LLM calls. Code is publicly available at https://github.com/manhitv/UAB.

Scaling test-time compute via extended reasoning has become a key paradigm for improving the capabilities of large language models (LLMs). However, existing approaches optimize reasoning under fixed or uniformly sampled token budgets, ignoring the fundamental mismatch between problem difficulty and allocated compute. This leads to overthinking on easy problems and underthinking on hard ones, resulting in suboptimal token efficiency across diverse reasoning scenarios. In this paper, we propose Budget-Adaptive Curriculum Reasoning (BCAE), a unified framework that jointly optimizes reasoning quality and token efficiency through three synergistic components: (1) a budget-conditioned unified policy that embeds the token budget as a continuous conditioning signal, eliminating the need for decoupled thinking and summarization strategies; (2) a curriculum-aware budget scheduler that adaptively shifts the training budget distribution from easy to hard problems based on real-time learning progress; and (3) a truncation-aware dense reward mechanism that provides fine-grained credit assignment at intermediate reasoning steps via process-level verification. We further introduce Budget-Conditioned Advantage Estimation (BCAE), a novel variance reduction technique that conditions the advantage baseline on the sampled budget, yielding more stable policy gradients. Experiments on mathematical reasoning benchmarks (MATH, GSM8K, AIME, and Minerva Math) demonstrate that BACR consistently outperforms other strong baselines across all token budgets, achieving up to 8.3% accuracy improvement under tight budgets while reducing average token consumption by 34% compared to unconstrained reasoning.

Large language models (LLMs) can achieve strong reasoning performance with sufficient computation, but they do not inherently know how much computation a task requires. We study budgeted inference-time reasoning for multiple problems under a strict global token constraint and formulate it as an Ordered Stochastic Multiple-Choice Knapsack Problem (OS-MCKP). This perspective highlights a meta-cognitive requirement---anticipating problem difficulty and allocating computation strategically. We propose ROI-Reasoning, a two-stage framework that endows LLMs with intrinsic, budget-aware rationality. First, Meta-Cognitive Fine-Tuning teaches models to predict reasoning cost levels and decide whether a problem is worth attempting before generation. Next, Rationality-Aware Reinforcement Learning optimizes sequential decision making under a hard token budget, allowing models to learn long-horizon allocation strategies. Experiments across mathematical reasoning, multiple-choice question answering, and code generation show that ROI-Reasoning improves scores and generally reduces reference-order regret under tight budgets, with up to a 417% relative score improvement over the strongest same-backbone baseline in mathematics.