Recommendations with Sparse Comparison Data: Provably Fast Convergence for Nonconvex Matrix Factorization

In this paper, we consider a recommender system that elicits user feedback through pairwise comparisons instead of ratings. We study the problem of learning personalised preferences from such comparison data via collaborative filtering. Similar to the classical matrix completion setting, we assume that users and items are endowed with low-dimensional latent features. These features give rise to user-item utilities, and the comparison outcomes are governed by a discrete choice model over these utilities. The task of learning these features is then formulated as a maximum likelihood problem over the comparison dataset. Despite the resulting optimization problem being nonconvex, we show that gradient-based methods converge exponentially to the latent features, given a warm start. Importantly, this result holds in a sparse data regime, where each user compares only a few pairs of items. Our main technical contribution is to extend key concentration results commonly used in matrix completion to our model. Simulations reveal that the empirical performance of the method exceeds theoretical predictions, even when some assumptions are relaxed. Our work demonstrates that learning personalised recommendations from comparison data is both computationally and statistically efficient.

Most recommendation systems, such as those used by Netflix or Amazon, ask users to rate items with stars or numbers. In this work, we explore an alternative approach where users express their preferences by choosing between two options, such as “Do you prefer A or B?” This type of feedback is often more natural and easier for users to provide. As with any recommendation system, the main challenge is to offer personalised suggestions for items a user has not yet seen.We adopt a well-established framework where the system builds hidden profiles for each user and item, which it uses to predict preferences. These profiles are learned by fitting model parameters to the observed data. Although this method works well in practice, it is difficult to analyse theoretically. The key challenges are that both user and item features are unknown (leading to nonconvexity), and each data point contains only binary comparison information (introducing nonlinearity).Our work is the first to provide theoretical guarantees that a simple and practical algorithm—gradient descent—can reliably learn personalised recommendations from comparison data. The main innovation lies in the careful use of classical probabilistic tools, in particular matrix concentration inequalities. Experiments support our findings, showing strong performance even beyond the idealised conditions assumed in the theory. These results suggest that learning from comparisons can be a practical and effective alternative to traditional rating-based recommendation systems.