Computation of Exact Bootstrap Confidence Intervals: Complexity and Deterministic Algorithms

AbstractThe bootstrap method is one of the major developments in statistics in the 20th century for computing confidence intervals directly from data. However, the bootstrap method is traditionally approximated with a randomized algorithm, which can sometimes produce inaccurate confidence intervals. In “Computation of Exact Bootstrap Confidence Intervals: Complexity and Deterministic Algorithms,” Bertsimas and Sturt present a new perspective of the bootstrap method through the lens of counting integer points in a polyhedron. Through this perspective, the authors develop the first computational complexity results and efficient deterministic approximation algorithm (fully polynomial time approximation scheme) for bootstrap confidence intervals, which unlike traditional methods, has guaranteed bounds on its error. In experiments on real and synthetic data sets from clinical trials, the proposed deterministic algorithms quickly produce reliable confidence intervals, which are significantly more accurate than those from randomization.The bootstrap is a nonparametric approach for calculating quantities, such as confidence intervals, directly from data. Since calculating exact bootstrap quantities is believed to be intractable, randomized resampling algorithms are traditionally used. In this paper, we present a new perspective on the bootstrap method through the lens of counting integer points in polyhedra. Through this new perspective, we make several advances for the bootstrap method, both theoretically and algorithmically. First, we establish several computational complexity results for the exact bootstrap method in the case of the sample mean. Second, we present the first efficient deterministic approximation algorithm (fully polynomial time approximation scheme) for producing exact bootstrap confidence intervals which, unlike traditional methods, has guaranteed bounds on the approximation error. Third, we develop a simple exact algorithm for exact bootstrap confidence intervals based on polynomial multiplication. We provide empirical evidence on real and synthetic data sets with several hundreds of data points that the proposed deterministic algorithms can quickly produce confidence intervals that are substantially more accurate than those from randomized methods, and thus are practical alternatives in applications such as clinical trials.