Contextual Search via Intrinsic Volumes
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We study the problem of contextual search, a multidimensional generalization of binary search that captures many problems in contextual decision-making. In contextual search, a learner is trying to learn the value of a hidden vector v ∈ [0,1]^d. Every round the learner is provided an adversarially-chosen context u_t ∈R^d, submits a guess p_t for the value of 〈 u_t, v〉, learns whether p_t < 〈 u_t, v〉, and incurs loss 〈 u_t, v〉. The learner's goal is to minimize their total loss over the course of T rounds. We present an algorithm for the contextual search problem for the symmetric loss function ℓ(θ, p) = |θ - p| that achieves O_d(1) total loss. We present a new algorithm for the dynamic pricing problem (which can be realized as a special case of the contextual search problem) that achieves O_d( T) total loss, improving on the previous best known upper bounds of O_d( T) and matching the known lower bounds (up to a polynomial dependence on d). Both algorithms make significant use of ideas from the field of integral geometry, most notably the notion of intrinsic volumes of a convex set. To the best of our knowledge this is the first application of intrinsic volumes to algorithm design.
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