Learning Preference Distributions From Distance Measurements

We introduce the problem of learning a distribution of user preferences over a set of items from noisy responses to distance queries. Rather than aiming to learn the preferences of each user, our goal is only to recover the overall distribution of user preferences. We show that distribution recovery can require just one response from each user. In contrast, learning the preferences of each user would require multiple responses from each user. Thus, learning preference distributions, rather than individual preferences, may be more practical in many applications. The preference distribution problem is formulated on a discrete domain in which items (e.g., products) and users' ideal preference points are located. We study both the noiseless and noisy settings in one dimension and provide sufficient conditions for identifiability of the underlying true distribution as a function of the set of items used for queries. We establish an upper bound on the total variation distance between the true distribution and the distribution learned via constrained least squares optimization problem for both noiseless and noisy settings. While the one-dimensional setting we consider is simple, our simulation results show that our proposed recovery technique extends to multidimensional settings and graph structures.