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Contextual Search via Intrinsic Volumes

59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pp. 268-282
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We study the problem of contextual search, a multidimensional generalization of bi- nary 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 ut ∈ Rd, submits a guess pt for the value of ⟨ut, v⟩, learns whether pt < ⟨ut, v⟩, and incurs loss ⟨ut, 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 Od(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 Od(log log T ) total loss, improving on the previous best known upper bounds of Od(logT) 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.