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

TL;DR

This paper analyzes a new recommender system learning problem using pairwise comparison data, demonstrating that gradient methods converge rapidly even with sparse data, underpinned by theoretical guarantees.

Contribution

It introduces a theoretical framework for learning user and item features from comparison data with provably fast convergence in nonconvex settings, even with sparse observations.

Findings

Gradient-based methods converge exponentially near the true solution.

The loss function exhibits restricted strong convexity in a nonconvex setting.

Efficient learning is possible from sparse comparison data.

Abstract

This paper provides a theoretical analysis of a new learning problem for recommender systems where users provide feedback by comparing pairs of items instead of rating them individually. We assume that comparisons stem from latent user and item features, which reduces the task of predicting preferences to learning these features from comparison data. Similar to the classical matrix factorization problem, the main challenge in this learning task is that the resulting loss function is nonconvex. Our analysis shows that the loss function exhibits (restricted) strong convexity near the true solution, which ensures gradient-based methods converge exponentially, given an appropriate 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 certain concentration inequalities commonly used in matrix completion to our model. Our work demonstrates that learning personalized recommendations from comparison data is computationally and statistically efficient.