广义二叉搜索的几何
The Geometry of Generalized Binary Search
摘要：
本文研究了通过一系列策略选择查询确定二进制值函数的问题。重点是一种称为广义二进制搜索（GBS）的算法。GBS是一个众所周知的贪婪算法，用于通过一系列策略选择的查询来确定二进制值函数。在每一步中，选择一个查询，最均匀地将所考虑的假设分成两个不相交的子集，这是经典二分法搜索的基本思想的自然推广。本文开发了新的不一致性和几何条件，在这些条件下，GBS实现了信息理论上的最优查询复杂度;即，给定N个假设的集合，GBS在不超过常数乘以log N个查询之后以正确的函数终止。此外，噪声容忍版本的GBS的开发，也达到了最佳的查询复杂度。这些结果适用于学习半空间，经常出现在图像处理和机器学习的问题。
英文摘要：
This paper investigates the problem of determining a binary-valued function through a sequence of strategically selected queries. The focus is an algorithm called Generalized Binary Search (GBS). GBS is a well-known greedy algorithm for determining a binary-valued function through a sequence of strategically selected queries. At each step, a query is selected that most evenly splits the hypotheses under consideration into two disjoint subsets, a natural generalization of the idea underlying classic binary search. This paper develops novel incoherence and geometric conditions under which GBS achieves the information-theoretically optimal query complexity; i.e., given a collection of N hypotheses, GBS terminates with the correct function after no more than a constant times log N queries. Furthermore, a noise-tolerant version of GBS is developed that also achieves the optimal query complexity. These results are applied to learning halfspaces, a problem arising routinely in image processing and machine learning.
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