Online Learning for Min Sum Set Cover and Pandora's Box
This work addresses stochastic optimization problems for online learning scenarios, offering a solution with theoretical guarantees but is incremental as it extends known offline results to online settings.
The paper tackles the problem of online learning for Min Sum Set Cover and Pandora's Box, where value vectors are presented online instead of from a known distribution, and presents a computationally efficient algorithm that is constant-competitive against the optimal search order.
Two central problems in Stochastic Optimization are Min Sum Set Cover and Pandora's Box. In Pandora's Box, we are presented with $n$ boxes, each containing an unknown value and the goal is to open the boxes in some order to minimize the sum of the search cost and the smallest value found. Given a distribution of value vectors, we are asked to identify a near-optimal search order. Min Sum Set Cover corresponds to the case where values are either 0 or infinity. In this work, we study the case where the value vectors are not drawn from a distribution but are presented to a learner in an online fashion. We present a computationally efficient algorithm that is constant-competitive against the cost of the optimal search order. We extend our results to a bandit setting where only the values of the boxes opened are revealed to the learner after every round. We also generalize our results to other commonly studied variants of Pandora's Box and Min Sum Set Cover that involve selecting more than a single value subject to a matroid constraint.