Reproduce “Recommendations with Sparse Comparison Data: Provably Fast Convergence for Nonconvex Matrix Factorization” on OpenResearch

About this paper

Suryanarayana Sankagiri, Jalal Etesami, Matthias Grossglauser

This work establishes the first theoretical recovery guarantees for nonconvex matrix factorization when applied to sparse pairwise comparison data in recommender systems. It demonstrates that projected gradient descent achieves exponential convergence to the true latent features under specific sparsity and initialization conditions.

Key insights

• Despite the inherent nonconvexity, the objective function behaves like a strongly convex and smooth function within a specific neighborhood of the true solution.
• Novel concentration results, derived using the Matrix Bernstein inequality, are essential for adapting theoretical matrix completion tools to the distinct sampling matrices of comparison data.
• Gradient-based methods can achieve provably fast, exponential convergence to the underlying preferences even when learning from sparse pairwise comparisons, given sufficient samples and a warm start.

Summary by alphaXiv.