  This work introduces the first small-loss and gradual-variation regret bounds
for online portfolio selection, marking the first instances of data-dependent
bounds for online convex optimization with non-Lipschitz, non-smooth losses.
The algorithms we propose exhibit sublinear regret rates in the worst cases and
achieve logarithmic regrets when the data is "easy," with per-iteration time
almost linear in the number of investment alternatives. The regret bounds are
derived using novel smoothness characterizations of the logarithmic loss, a
local norm-based analysis of following the regularized leader (FTRL) with
self-concordant regularizers, which are not necessarily barriers, and an
implicit variant of optimistic FTRL with the log-barrier.
