Generalized Binary Search For Split-Neighborly Problems

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.