Contextual Pandora's Box
This work addresses the problem of efficient decision-making under uncertainty with contextual information for researchers in stochastic optimization and online learning, representing an incremental extension by incorporating context into the online Pandora's Box setting.
The authors tackled the online contextual Pandora's Box problem, where alternatives have unknown distributions that change each round, and developed a no-regret algorithm that performs comparably to an optimal algorithm with full knowledge of the distributions, even in the bandit setting.
Pandora's Box is a fundamental stochastic optimization problem, where the decision-maker must find a good alternative while minimizing the search cost of exploring the value of each alternative. In the original formulation, it is assumed that accurate distributions are given for the values of all the alternatives, while recent work studies the online variant of Pandora's Box where the distributions are originally unknown. In this work, we study Pandora's Box in the online setting, while incorporating context. At every round, we are presented with a number of alternatives each having a context, an exploration cost and an unknown value drawn from an unknown distribution that may change at every round. Our main result is a no-regret algorithm that performs comparably well to the optimal algorithm which knows all prior distributions exactly. Our algorithm works even in the bandit setting where the algorithm never learns the values of the alternatives that were not explored. The key technique that enables our result is a novel modification of the realizability condition in contextual bandits that connects a context to a sufficient statistic of each alternative's distribution (its "reservation value") rather than its mean.