Multidimensional Binary Search for Contextual Decision-Making

TL;DR

This paper introduces a polynomial-time algorithm for multidimensional binary search in contextual decision-making, achieving near-optimal regret bounds by combining volume cutting and a novel cylindrification technique.

Contribution

The paper presents a new algorithm called Projected Volume that efficiently solves multidimensional search problems with near-optimal regret bounds, introducing the cylindrification geometric technique.

Findings

Achieves regret $O(d ext{log}(d/ extepsilon))$, nearly optimal.

Introduces the cylindrification geometric technique.

Provides a polynomial-time algorithm for the problem.

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 $\epsilon$ away from the true answer. We construct a polynomial time algorithm that we call Projected Volume achieving regret $O(d\log(d/\epsilon))$, 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.