  We study the decades-old problem of online portfolio management and propose
the first algorithm with logarithmic regret that is not based on Cover's
Universal Portfolio algorithm and admits much faster implementation.
Specifically Universal Portfolio enjoys optimal regret $\mathcal{O}(N\ln T)$
for $N$ financial instruments over $T$ rounds, but requires log-concave
sampling and has a large polynomial running time. Our algorithm, on the other
hand, ensures a slightly larger but still logarithmic regret of
$\mathcal{O}(N^2(\ln T)^4)$, and is based on the well-studied Online Mirror
Descent framework with a novel regularizer that can be implemented via standard
optimization methods in time $\mathcal{O}(TN^{2.5})$ per round. The regret of
all other existing works is either polynomial in $T$ or has a potentially
unbounded factor such as the inverse of the smallest price relative.
