A generalised log-determinant regularizer for online semi-definite programming and its applications

We consider a variant of online semi-definite programming problem (OSDP): The decision space consists of semi-definite matrices with bounded $\bm{\Gamma}$-trace norm, which is a generalization of trace norm defined by a positive definite matrix $\Gamma.$ To solve this problem, we utilise the follow-the-regularized-leader algorithm with a $\Gamma$-dependent log-determinant regularizer. Then we apply our generalised setting and our proposed algorithm to online matrix completion(OMC) and online similarity prediction with side information. In particular, we reduce the online matrix completion problem to the generalised OSDP problem, and the side information is represented as the $\Gamma$ matrix. Hence, due to our regret bound for the generalised OSDP, we obtain an optimal mistake bound for the OMC by removing the logarithmic factor.