The Sample Complexity of Best-$k$ Items Selection from Pairwise Comparisons
This work addresses the sample efficiency challenge in active learning for ranking and selection tasks, which is incremental as it builds on existing stochastic comparison models to derive tighter bounds.
This paper tackles the problem of selecting the best-k items from pairwise comparisons with noisy results, establishing sample complexity bounds for both probably approximately correct (PAC) and exact selection under strong stochastic transitivity and stochastic triangle inequality. It provides algorithms with upper bounds matching lower bounds up to constant or logarithmic factors, such as an optimal algorithm for PAC selection and near-optimal ones for exact selection.
This paper studies the sample complexity (aka number of comparisons) bounds for the active best-$k$ items selection from pairwise comparisons. From a given set of items, the learner can make pairwise comparisons on every pair of items, and each comparison returns an independent noisy result about the preferred item. At any time, the learner can adaptively choose a pair of items to compare according to past observations (i.e., active learning). The learner's goal is to find the (approximately) best-$k$ items with a given confidence, while trying to use as few comparisons as possible. In this paper, we study two problems: (i) finding the probably approximately correct (PAC) best-$k$ items and (ii) finding the exact best-$k$ items, both under strong stochastic transitivity and stochastic triangle inequality. For PAC best-$k$ items selection, we first show a lower bound and then propose an algorithm whose sample complexity upper bound matches the lower bound up to a constant factor. For the exact best-$k$ items selection, we first prove a worst-instance lower bound. We then propose two algorithms based on our PAC best items selection algorithms: one works for $k=1$ and is sample complexity optimal up to a loglog factor, and the other works for all values of $k$ and is sample complexity optimal up to a log factor.