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

arXiv: https://arxiv.org/abs/2502.20033 · HTML: https://mlsanity.com/paper/2502.20033/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 26 references that are in the map.
- First posted on arXiv 27 Feb 2025.

## Builds on (most cited first, 12 of 12)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysis](https://mlsanity.com/paper/1704.00708/) (2017)
- [Matrix Completion has No Spurious Local Minimum](https://mlsanity.com/paper/1605.07272/) (2016)
- [Preference-based Online Learning with Dueling Bandits: A Survey](https://mlsanity.com/paper/1807.11398/) (2018)
- [Guaranteed Matrix Completion via Non-convex Factorization](https://mlsanity.com/paper/1411.8003/) (2014)
- [Fast low-rank estimation by projected gradient descent: General statistical and algorithmic guarantees](https://mlsanity.com/paper/1509.03025/) (2015)
- [Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution](https://mlsanity.com/paper/1711.10467/) (2017)
- [Estimation from Pairwise Comparisons: Sharp Minimax Bounds with Topology Dependence](https://mlsanity.com/paper/1505.01462/) (2015)
- [Convergence Analysis for Rectangular Matrix Completion Using Burer-Monteiro Factorization and Gradient Descent](https://mlsanity.com/paper/1605.07051/) (2016)
- [Stochastic Contextual Dueling Bandits under Linear Stochastic Transitivity Models](https://mlsanity.com/paper/2202.04593/) (2022)
- [Learning from Comparisons and Choices](https://mlsanity.com/paper/1704.07228/) (2017)
- [Preference Completion: Large-scale Collaborative Ranking from Pairwise Comparisons](https://mlsanity.com/paper/1507.04457/) (2015)

## Built on by (most cited first, 1 of 1)

- [Graph Learning Is Suboptimal in Causal Bandits](https://mlsanity.com/paper/2510.16811/) (2025)

## Abstract (arXiv)

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.