Contextual Pandora's Box

TL;DR

This paper introduces a no-regret algorithm for the online contextual Pandora's Box problem, effectively handling unknown, changing distributions and bandit feedback by linking contexts to reservation values.

Contribution

It presents a novel approach that extends Pandora's Box to the online, contextual setting with unknown distributions, achieving no-regret performance.

Findings

Developed a no-regret algorithm for online contextual Pandora's Box.

The algorithm performs well even with bandit feedback.

Introduced a new realizability condition linking context to reservation values.

Abstract

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.