Motivated by an application of eliciting users' preferences, we investigate the problem of learning hemimetrics, i.e., pairwise distances among a set of $n$ items that satisfy triangle inequalities and non-negativity constraints. In our application, the (asymmetric) distances quantify private costs a user incurs when substituting one item by another. We aim to learn these distances (costs) by asking the users whether they are willing to switch from one item to another for a given incentive offer. Without exploiting structural constraints of the hemimetric polytope, learning the distances between each pair of items requires $\Theta(n^2)$ queries. We propose an active learning algorithm that substantially reduces this sample complexity by exploiting the structural constraints on the version space of hemimetrics. Our proposed algorithm achieves provably-optimal sample complexity for various instances of the task. For example, when the items are embedded into $K$ tight clusters, the sample complexity of our algorithm reduces to $O(n K)$. Extensive experiments on a restaurant recommendation data set support the conclusions of our theoretical analysis.

@inproceedings{singla16hemimetric,
	author = {Adish Singla and Sebastian Tschiatschek and Andreas Krause},
	booktitle = {Proc. International Conference on Machine Learning (ICML)},
	month = {June},
	title = {Actively Learning Hemimetrics with Applications to Eliciting User Preferences},
	year = 2016}