  We consider a variant of online semi-definite programming problem (OSDP): The
decision space consists of semi-definite matrices with bounded $\Gamma$-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.
