Probably Approximately Correct Greedy Maximization

Submodular function maximization finds application in a variety of real-world decision-making problems. However, most existing methods, based on greedy maximization, assume it is computationally feasible to evaluate F, the function being maximized. Unfortunately, in many realistic settings F is too expensive to evaluate exactly even once. We present probably approximately correct greedy maximization, which requires access only to cheap anytime confidence bounds on F and uses them to prune elements. We show that, with high probability, our method returns an approximately optimal set. We propose novel, cheap confidence bounds for conditional entropy, which appears in many common choices of F and for which it is difficult to find unbiased or bounded estimates. Finally, results on a real-world dataset from a multi-camera tracking system in a shopping mall demonstrate that our approach performs comparably to existing methods, but at a fraction of the computational cost.