Online matrix prediction for sparse loss matrices

We consider an online matrix prediction problem. FTRL is a standard method to deal with online prediction tasks, which makes predictions by minimizing the cumulative loss function and the regularizer function. There are three popular regularizer functions for matrices, Frobenius norm, negative entropy and log-determinant. We propose an FTRL based algorithm with log-determinant as the regularizer and show a regret bound of the algorithm. Our main contribution is to show that the log-determinant regularization is effective when loss matrices are sparse. We also show that our algorithm is optimal for the online collaborative filtering problem with the log-determinant regularization.