  We study online convex optimization in a setting where the learner seeks to
minimize the sum of a per-round hitting cost and a movement cost which is
incurred when changing decisions between rounds. We prove a new lower bound on
the competitive ratio of any online algorithm in the setting where the costs
are $m$-strongly convex and the movement costs are the squared $\ell_2$ norm.
This lower bound shows that no algorithm can achieve a competitive ratio that
is $o(m^{-1/2})$ as $m$ tends to zero. No existing algorithms have competitive
ratios matching this bound, and we show that the state-of-the-art algorithm,
Online Balanced Decent (OBD), has a competitive ratio that is
$\Omega(m^{-2/3})$. We additionally propose two new algorithms, Greedy OBD
(G-OBD) and Regularized OBD (R-OBD) and prove that both algorithms have an
$O(m^{-1/2})$ competitive ratio. The result for G-OBD holds when the hitting
costs are quasiconvex and the movement costs are the squared $\ell_2$ norm,
while the result for R-OBD holds when the hitting costs are $m$-strongly convex
and the movement costs are Bregman Divergences. Further, we show that R-OBD
simultaneously achieves constant, dimension-free competitive ratio and
sublinear regret when hitting costs are strongly convex.
