Globally-Optimal Greedy Active Sequential Estimation

Abstract

Motivated by modern applications such as computerized adaptive testing, sequential rank aggregation, and heterogeneous data source selection, we study the problem of active sequential estimation. The goal is to design an adaptive experiment selection rule and an estimator for more accurate parameter estimation. Greedy information-based experiment selection rules, which optimize information gain one step ahead, have been employed in practice thanks to their computational convenience, flexibility to context or task changes, and broad applicability. However, the optimality of greedy methods under a sequential decision theory framework is only established in the one-dimensional case, partly due to the problem’s combinatorial nature and the seemingly limited capacity of greedy algorithms. In this study, we close the gap for multidimensional problems. We cast the problem under a sequential decision theory framework with generalized risk measures for a large class of design-and-estimation methods. We propose adopting the maximum likelihood estimator with a class of greedy experiment selection rules. This class encompasses both existing methods and introduces new methods with improved numerical efficiency. We prove that these methods achieve asymptotic optimality when the risk measure aligns with the selection rule. Additionally, we establish that the proposed estimators are consistent and asymptotically normal, and further extend the results to allow early stopping rules. We also perform extensive numerical studies on both simulated and real data to illustrate the efficacy of the proposed methods.