Open sourcing QMF for matrix factorization

Since the objective is an expectation, we can use the Stochastic Gradient Descent (SGD) algorithm to minimize it, by iteratively minimizing the loss on sampled triplets of (user, positive item, negative item). Our implementation uses a form of parallel and asynchronous SGD updates (called Hogwild! [3]), which can give near-linear speedups in terms of the number of processors, especially when the data is sparse, so that concurrent updates are unlikely to be on the same parameters and overwrite each other. In the case of recommendation problems, the data is usually very sparse (unless, e.g., there are few items and most users have seen the same items), and this approach works very well.