Pandora's Box with Correlations: Learning and Approximation
In the Pandora's Box problem, the algorithm is provided with a number of boxes with unknown (stochastic) rewards contained inside them. The algorithm can open any box at some cost, discover the reward inside, and based on these observations can choose one box and keep the reward contained in it. Given the distributions from which the rewards are drawn, the algorithm must determine an order in which to open the boxes as well as when to stop and accept the best reward found so far. In general, an optimal algorithm may make both decisions adaptively based on instantiations observed previously. The Pandora's Box problem and its extensions capture many kinds of optimization problems with stochastic input where the algorithm can obtain instantiations of input random variables at some cost. Previous work on these problems assumes that different random variables in the input are distributed independently. As such it does not capture many real-world settings. In this work, we provide the first algorithms for Pandora's Box-type problems with correlations. In the independent setting, optimal algorithms are non-adaptive and based on the notion of the Gittins index. These techniques fail to extend to the correlated case. We assume that the algorithm has access to samples drawn from the joint distribution on input and provide solutions that require few samples; are computationally efficient; and guarantee approximate optimality.
This is joint work with Evangelia Gergatsouli, Yifeng Teng, Christos Tzamos, and Ruimin Zhang.