发表机构
Matrosov Institute for System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences; Moscow Institute of Physics and Technology; Innopolis University（俄罗斯科学院西伯利亚分院马特罗索夫系统动力学与控制理论研究所; 莫斯科物理技术学院; 因诺波利斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究分析贪心盒子分割在符号预言机模型下的安全性，确定减半为极小极大最优，给出尖锐阈值与收敛条件，并区分比较与符号查询的复杂度。
AI 中文摘要
贪心盒子分割在中心读取梯度符号并保留一个较小的盒子。其可累加的旅行预算使得错误的排除变得不可逆。我们确定了这种几何结构何时是安全的。在坐标方向符号一致且不必凸的类别上，减半在符号向量预言机模型中是极小极大最优的。一个无量纲的符号缺陷产生一个尖锐的标量包络，由一个光滑凸势函数达到；一个三维轨迹最终距离目标比其初始半径更远。对于二次型，当加权绝对行缺陷至多为 $(2 \beta - 1)$ 时，具有纵横比 $w$ 和收缩率 $\beta$ 的普适收敛恰好成立。Perron 权重使该缺陷最小化。自适应半尺寸避免了初始扩大，并在持续误差下收敛到精确的带帽不动点极限。对于二元比较，独立探测比例 $q$ 在共同目标单调部分上恰好当 $(q \leq 2 \beta - 1)$ 时允许普适定位；耦合探测给出尖锐阈值 $2/3$。匹配的信息界区分了比较与符号向量查询。误差带给出有限时间保证和尖锐的残差下限。扩展和可复现实验确定了冻结、平滑、投票、多重启动和随机矩阵证书的假设与成本。
英文摘要
Greedy box splitting reads gradient signs at the centre and retains one smaller box. Its summable travel budget makes an incorrect exclusion irreversible. We determine when this geometry is safe. Halving is minimax optimal in the sign-vector oracle model on a coordinatewise sign-consistent class that need not be convex. A dimensionless sign defect yields a sharp scalar envelope, attained by one smooth convex potential; a three-dimensional trajectory ends farther from the target than its initial radius. For quadratics, universal convergence with aspect $w$ and contraction beta holds exactly when the weighted absolute row defect is at most $(2 β- 1)$. Perron weights minimise this defect. Adaptive half-sizes avoid initial enlargement and, under persistent error, converge to an exact capped fixed-point limit. For binary comparisons, an independent probe fraction $q$ permits universal localisation on common-target monotone sections exactly when $(q \leq 2 β- 1)$; coupled probes give the sharp threshold $2/3$. Matching information bounds distinguish comparisons from sign-vector queries. Error bands give finite-time guarantees and sharp residual floors. Extensions and reproducible experiments identify the assumptions and costs of freezing, smoothing, voting, multistart and random-matrix certificates.
