  We revisit the classic online portfolio selection problem, where at each
round a learner selects a distribution over a set of portfolios to allocate its
wealth. It is known that for this problem a logarithmic regret with respect to
Cover's loss is achievable using the Universal Portfolio Selection algorithm,
for example. However, all existing algorithms that achieve a logarithmic regret
for this problem have per-round time and space complexities that scale
polynomially with the total number of rounds, making them impractical. In this
paper, we build on the recent work by Haipeng et al. 2018 and present the first
practical online portfolio selection algorithm with a logarithmic regret and
whose per-round time and space complexities depend only logarithmically on the
horizon. Behind our approach are two key technical novelties of independent
interest. We first show that the Damped Online Newton steps can approximate
mirror descent iterates well, even when dealing with time-varying regularizers.
Second, we present a new meta-algorithm that achieves an adaptive logarithmic
regret (i.e. a logarithmic regret on any sub-interval) for mixable losses.
