用少量强纠错者引导弱智能体群达成正确共识，优化成本与位置。
How Much Does Correctness Cost? Budgeted Placement of Strong Correctors in a Weak Multi-Agent Swarm
- 基于图结构建模，通过代价耦合的凹函数调节纠错者强度。
- 证明一致性指标具次模性，贪心算法在任意预算下近似最优。
- 任务特性决定投入策略：数学验证需少而强，代码追踪宜多而平。
廉价且不可靠的智能体群可通过少数昂贵的‘预言者’纠错器引导至正确共识。本文研究在给定预算下应投入多少、如何布局这些纠错器。将群体共识建模为图上的状态同步过程，每个预言者以代价耦合的凹函数强度固定一个节点向真值逼近，并以协方差矩阵逆的迹 H(R)=tr M(R)^{-1} 衡量一致性质量。理论证明：即便预言者强度不同，H(R) 仍保持次模性，因此成本效益贪心算法可在任意预算下达到最优解的 1-1/e 近似。反推可得预算-精度边界 B*(eps)，即保证 eps 精度所需的最低支出：在完全图上可得闭式解；当所有预言者成本相同时，最小预言者数为 k*。是否选择少数强或多数中等强度的预言者，取决于代价-质量曲线的曲率——递减回报倾向分散部署。在 Qwen3 阶梯（0.6-32B）上测试显示，数学验证任务呈凹型，代码追溯任务则呈凸型，结论真实依赖任务类型。
原文摘要 · Abstract (English)
A cheap swarm of unreliable agents can be steered to a correct consensus by a few strong, expensive "oracle" correctors. We ask how much one must spend, and where to place the oracles. We model the swarm as a consensus on a graph in which each oracle pins one node toward the truth at a cost-coupled, concave strength, and measure quality by the coherence H(R)=tr M(R)^{-1}. Our first result is that H stays submodular (each added oracle helps less than the last) even when the oracles differ in strength, so a cost-benefit greedy comes within 1-1/e of the best placement at any budget. Inverting the budget gives the budget-correctness frontier B*(eps), the least spend that guarantees an eps-correct consensus: closed-form on the complete graph, and a minimal oracle count k* when oracles cost the same. Whether a budget then buys a few strong oracles or many medium onese curvature of the cost-quality law: diminishing returns favour spreadsharply increasion. Measured onthe Qwen3 ladder (0.6-32B), the law is concave for math verificatio convex foremergent code tracing, so the verdict is genuinely task-dependent.https://github.com/YehudaItkin/budgeted-oracle-placemen
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