Bayesian Low-Rank Determinantal Point Processes

Abstract

Determinantal point processes (DPPs) are an emerging model for encoding probabilities over subsets, such as shopping baskets, selected from a ground set, such as an item catalog. They have recently proved to be appealing models for a number of machine learning tasks, including product recommendation. DPPs are parametrized by a positive semi-definite kernel matrix. Prior work has shown that using a low-rank factorization of this kernel provides scalability improvements that open the door to training on large-scale datasets and computing online recommendations, both of which are infeasible with standard DPP models that use a full-rank kernel. A low-rank DPP model can be trained using an optimization-based method, such as stochastic gradient ascent, to find a point estimate of the kernel parameters, which can be performed efficiently on large-scale datasets. However, this approach requires careful tuning of regularization parameters to prevent overfitting and provide good predictive performance, which can be computationally expensive. In this paper we present a Bayesian method for learning a low-rank factorization of this kernel, which provides automatic control of regularization. We show that our Bayesian low-rank DPP model can be trained efficiently using stochastic gradient Hamiltonian Monte Carlo (SGHMC). Our Bayesian model generally provides better predictive performance on several real-world product recommendation datasets than optimization-based low-rank DPP models trained using stochastic gradient ascent, and better performance than several state-of-the art recommendation methods in many cases.