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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(dlog(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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