Motivated by an application of eliciting users’preferences, we investigate the problem of learn-ing hemimetrics,i.e., pairwise distances among aset ofnitems that satisfy triangle inequalities andnon-negativity constraints. In our application, the(asymmetric) distances quantify private costs auser incurs when substituting one item by another.We aim to learn these distances (costs) by askingthe users whether they are willing to switch fromone item to another for a given incentive offer.Without exploiting structural constraints of thehemimetric polytope, learning the distances be-tween each pair of items requiresΘ(n2)queries.We propose an active learning algorithm thatsubstantially reduces this sample complexity byexploiting the structural constraints on the versionspace of hemimetrics. Our proposed algorithmachieves provably-optimal sample complexity forvarious instances of the task. For example, whenthe items are embedded intoKtight clusters, thesample complexity of our algorithm reduces toO(nK). Extensive experiments on a restaurantrecommendation data set support the conclusionsof our theoretical analysis.

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