1-bit Matrix Completion: PAC-Bayesian Analysis of a Variational Approximation

Abstract: Due to challenging applications such as collaborative filtering, the matrix completion problem has been widely studied in the past few years. Different approaches rely on different structure assumptions on the matrix in hand. Here, we focus on the completion of a (possibly) low-rank matrix with binary entries, the so-called 1-bit matrix completion problem. Our approach relies on tools from machine learning theory: empirical risk minimization and its convex relaxations. We propose an algorithm to compute a variational approximation of the pseudo-posterior. Thanks to the convex relaxation, the corresponding minimization problem is bi-convex, and thus the method behaves well in practice. We also study the performance of this variational approximation through PAC-Bayesian learning bounds. On the contrary to previous works that focused on upper bounds on the estimation error of M with various matrix norms, we are able to derive from this analysis a PAC bound on the prediction error of our algorithm. We focus essentially on convex relaxation through the hinge loss, for which we present the complete analysis, a complete simulation study and a test on the MovieLens data set. However, we also discuss a variational approximation to deal with the logistic loss.