On the Optimality of Tracking Fisher Information in Adaptive Testing with Stochastic Binary Responses

Abstract

We study the problem of estimating a continuous ability parameter from sequential binary responses by actively asking questions with varying difficulties, a setting that arises naturally in adaptive testing and online preference learning. Our goal is to certify that the estimate lies within a desired margin of error, using as few queries as possible. We propose a simple algorithm that adaptively selects questions to maximize Fisher information and updates the estimate using a method-of-moments approach, paired with a novel test statistic to decide when the estimate is accurate enough. We prove that this Fisher-tracking strategy achieves optimal performance in both fixed-confidence and fixed-budget regimes, which are commonly invested in the best-arm identification literature. Our analysis overcomes a key technical challenge in the fixed-budget setting -- handling the dependence between the evolving estimate and the query distribution -- by exploiting a structural symmetry in the model and combining large deviation tools with Ville's inequality. Our results provide rigorous theoretical support for simple and efficient adaptive testing procedures.