Estimating the Average of a Lipschitz-Continuous Function from One Sample

Abstract: We study the problem of estimating the average of a Lipschitz continuous function $f$ defined over a metric space, by querying $f$ at only a single point. More specifically, we explore the role of randomness in drawing this sample. Our goal is to find a distribution minimizing the expected estimation error against an adversarially chosen Lipschitz continuous function. Our work falls into the broad class of estimating aggregate statistics of a function from a small number of carefully chosen samples. The general problem has a wide range of practical applications in areas as diverse as sensor networks, social sciences and numerical analysis. However, traditional work in numerical analysis has focused on asymptotic bounds, whereas we are interested in the \emph{best} algorithm. For arbitrary discrete metric spaces of bounded doubling dimension, we obtain a PTAS for this problem. In the special case when the points lie on a line, the running time improves to an FPTAS. Both algorithms are based on approximately solving a linear program with an infinite set of constraints, by using an approximate separation oracle. For Lipschitz-continuous functions over $[0,1]$, we calculate the precise achievable error as $1-\frac{\sqrt{3}}{2} \approx 0.134$, which improves upon the \quarter which is best possible for deterministic algorithms.