Bisection Search with Noisy Responses
Rolf Waeber, Peter I. Frazier, Shane G. Henderson · SIAM Journal on Control and Optimization · 2013
Bisection search is the most efficient algorithm for locating a unique point $X^{\ast} \in [0,1]$ when we are able to query an oracle only about whether $X^{\ast}$ lies to the left or right of a point $x$ of our choosing. We study a noisy version of this classic problem, where the oracle's response is correct only with probability $p$. The probabilistic bisection algorithm (PBA) introduced by Horstein [IEEE Trans. Inform. Theory, 9 (1963), pp. 136--143] can be used to locate $X^{\ast}$ in this setting. While the method works extremely well in practice, very little is known about its theoretical properties. In this paper, we provide several key findings about the PBA, which lead to the main conclusion that the expected absolute residuals of successive search results, i.e., $\mathbb{E}[|X^{\ast} - X_n|]$, converge to 0 at a geometric rate.