Dynamic K-Head: Difficulty-Supervised Budget Selection for Continuous Latent Reasoning

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.

Long chain-of-thought~(CoT) reasoning improves the problem-solving capability of large reasoning models~(LRMs), but often incurs substantial inference cost and overthinking. Existing efficiency methods primarily rely on prompting, post-hoc compression, or length-regularized optimization; these approaches can be sensitive to model scale, depend on external estimators, or under-allocate reasoning to difficult instances. We propose Draft-Thinking, a lightweight adaptation framework that learns a model-adapted draft CoT: a compact, self-contained working derivation that directly supports the final answer. We identify a Cross-Scale Draft Mismatch: concise traces generated by a strong teacher can be too sparse for a smaller target model. To address this mismatch, Progressive Draft Adaptation first transfers a concise reasoning-form prior through supervised fine-tuning (SFT) and then applies two-stage reinforcement learning (RL) with progressively harder data and relaxed response budgets to calibrate the reasoning granularity to the target model. The method requires neither post-hoc trajectory compression nor an explicit length-penalty reward. At inference time, draft, adaptive, and step-by-step prompts provide efficiency-oriented, balanced, and detailed reasoning modes, respectively. The step-by-step mode largely preserves the original model's accuracy while using fewer tokens. With only 847 training problems and 120 RL optimization steps, Draft-Thinking substantially reduces reasoning cost across six mathematical and scientific reasoning benchmarks. On MATH500, its draft mode reduces the average response length from 5,668 to 986 tokens, an 82.6% reduction, while achieving 90.6% accuracy compared with 93.0% for the original long CoT model.