Generalized Binary Search For Split-Neighborly Problems

TL;DR

This paper introduces the split-neighborly condition, a weaker requirement than k-neighborliness, enabling generalized binary search to achieve optimal logarithmic query complexity in certain hypothesis testing problems.

Contribution

It defines the split-neighborly condition and proves that four problems not k-neighborly are split-neighborly, leading to optimal query costs.

Findings

Split-neighborly condition is weaker than k-neighborly.

Four problems are split-neighborly despite not being k-neighborly.

Achieves O(log n) query complexity for these problems.

Abstract

In sequential hypothesis testing, Generalized Binary Search (GBS) greedily chooses the test with the highest information gain at each step. It is known that GBS obtains the gold standard query cost of $O(\log n)$ for problems satisfying the $k$-neighborly condition, which requires any two tests to be connected by a sequence of tests where neighboring tests disagree on at most $k$ hypotheses. In this paper, we introduce a weaker condition, split-neighborly, which requires that for the set of hypotheses two neighbors disagree on, any subset is splittable by some test. For four problems that are not $k$-neighborly for any constant $k$, we prove that they are split-neighborly, which allows us to obtain the optimal $O(\log n)$ worst-case query cost.