CALIBUDGET: Calibration-Guided Source Allocation for Fixed-Budget Mixed-Reasoning Adaptation

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.

Post-training LLM pruning relies on a small calibration set, usually chosen heuristically and judged by WikiText perplexity or by mean accuracy on general-domain commonsense benchmarks. This single-axis view hides capability-specific effects: at 60% sparsity, strategies separated by $2.85$ points in averaged commonsense accuracy diverge by $51.9$ points in code retention, and across 15 sources no single one preserves all capabilities. We treat calibration as a capability-coverage problem. Balanced, a uniform four-way mixture under a fixed budget, tests the principle on real corpora: it outperforms every evaluated single source, and its margin over C4 grows from $+4.6$ points at 30% sparsity to $+18.8$ at 60%. The advantage holds across model scale and a second model family, and transfers to held-out benchmarks disjoint from calibration construction. When aligned corpora are unavailable, IGSP approximates that mixture from the base model and the evaluation taxonomy alone, outperforming prior self-generation methods by $7.23$--$12.79$ points on the LLaMA main suite. On Qwen2.5-7B, IGSP exceeds Balanced on both the main suite and held-out benchmarks. Capability coverage is a first-order design variable for high-sparsity pruning.

Evaluating large language models increasingly relies on LLM-as-a-judge protocols, but such evaluations remain costly: different judges have different prices and reliabilities, and the difficulty of each prompt-response pair can vary substantially. This raises a basic allocation question: under a fixed budget, how should one distribute evaluation queries across heterogeneous judges and instances to obtain the most accurate score estimates? We formalize this question as budgeted heteroskedastic multi-judge estimation. Given $K$ prompt-response pairs, $J$ judges with known costs, and unknown query-judge variances, the goal is to estimate a bounded score vector while minimizing an $\ell_p$-error. Our first contribution is to analyze the inverse-variance weighted estimator (IVWE) and to derive the oracle allocation that minimizes its error rate. Since this allocation depends on the unknown variances, we then address the practical unknown-variance setting by proposing EST-IVWE, an adaptive algorithm that constructs and leverages optimistically biased variance estimates to stabilize the empirical allocation. We prove that EST-IVWE matches the oracle IVWE rate up to lower-order terms in the budget. Our second and central theoretical contribution is a matching local minimax lower bound, which establishes the instance-optimality of the proposed algorithms. A key technical insight is that Fano-type high-probability arguments are too coarse for this problem: their packing construction loses the local variance structure that governs the optimal allocation. We instead use an Assouad-type in-expectation argument, based on local perturbations, which preserves this structure and yields the sharp allocation-dependent lower bound. Finally, we numerically validate the superiority of our approach over naïve uniform allocation on synthetic and HelpSteer2 datasets.