  We consider online linear optimization over symmetric positive semi-definite
matrices, which has various applications including the online collaborative
filtering. The problem is formulated as a repeated game between the algorithm
and the adversary, where in each round t the algorithm and the adversary choose
matrices X_t and L_t, respectively, and then the algorithm suffers a loss given
by the Frobenius inner product of X_t and L_t. The goal of the algorithm is to
minimize the cumulative loss. We can employ a standard framework called Follow
the Regularized Leader (FTRL) for designing algorithms, where we need to choose
an appropriate regularization function to obtain a good performance guarantee.
We show that the log-determinant regularization works better than other popular
regularization functions in the case where the loss matrices L_t are all
sparse. Using this property, we show that our algorithm achieves an optimal
performance guarantee for the online collaborative filtering. The technical
contribution of the paper is to develop a new technique of deriving performance
bounds by exploiting the property of strong convexity of the log-determinant
with respect to the loss matrices, while in the previous analysis the strong
convexity is defined with respect to a norm. Intuitively, skipping the norm
analysis results in the improved bound. Moreover, we apply our method to online
linear optimization over vectors and show that the FTRL with the Burg entropy
regularizer, which is the analogue of the log-determinant regularizer in the
vector case, works well.
