This paper considers the problem of learning the ranking of a set of stochastic alternatives based upon incomplete information (i.e., a limited number of samples). We describe a system that, at each decision cycle, outputs either a complete ordering on the hypotheses or decides to gather additional information (i.e., observations) at some cost. The ranking problem is a generalization of the previously studied hypothesis selection problem in selection, an algorithm must select the single best hypothesis, while in ranking, an algorithm must order all the hypotheses. The central problem we address is achieving the desired ranking quality while minimizing the cost of acquiring additional samples. We describe two algorithms for hypothesis ranking and their application for the probably approximately correct (PAC) and expected loss (EL) learning criteria. Empirical results are provided to demonstrate the effectiveness of these ranking procedures on both synthetic and real-world datasets.
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