Discriminative Learning via Adaptive Questioning

We consider the problem of designing an adaptive sequence of questions that optimally classify a candidate's ability into one of several categories or discriminative grades. A candidate's ability is modeled as an unknown parameter, which, together with the difficulty of the question asked, determines the likelihood with which s/he is able to answer a question correctly. The learning algorithm is only able to observe these noisy responses to its queries. We consider this problem from a fixed confidence-based $\delta$-correct framework, that in our setting seeks to arrive at the correct ability discrimination at the fastest possible rate while guaranteeing that the probability of error is less than a pre-specified and small $\delta$. In this setting we develop lower bounds on any sequential questioning strategy and develop geometrical insights into the problem structure both from primal and dual formulation. In addition, we arrive at algorithms that essentially match these lower bounds. Our key conclusions are that, asymptotically, any candidate needs to be asked questions at most at two (candidate ability-specific) levels, although, in a reasonably general framework, questions need to be asked only at a single level. Further, and interestingly, the problem structure facilitates endogenous exploration, so there is no need for a separately designed exploration stage in the algorithm.