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arXiv · 1611.00829

Multidimensional Binary Search for Contextual Decision-Making

Ilan Lobel·Renato Paes Leme·Adrian Vladu
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Abstract

We consider a multidimensional search problem that is motivated by questions in contextual decision-making, such as dynamic pricing and personalized medicine. Nature selects a state from a $d$-dimensional unit ball and then generates a sequence of $d$-dimensional directions. We are given access to the directions, but not access to the state. After receiving a direction, we have to guess the value of the dot product between the state and the direction. Our goal is to minimize the number of times when our guess is more than $ε$ away from the true answer. We construct a polynomial time algorithm that we call Projected Volume achieving regret $O(d\log(d/ε))$, which is optimal up to a $\log d$ factor. The algorithm combines a volume cutting strategy with a new geometric technique that we call cylindrification.

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Ilan Lobel, Renato Paes Leme, Adrian Vladu. 2017-04-26. Multidimensional Binary Search for Contextual Decision-Making. https://arxiv.org/abs/1611.00829

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