  Online mirror descent (OMD) and dual averaging (DA) -- two fundamental
algorithms for online convex optimization -- are known to have very similar
(and sometimes identical) performance guarantees when used with a fixed
learning rate. Under dynamic learning rates, however, OMD is provably inferior
to DA and suffers a linear regret, even in common settings such as prediction
with expert advice. We modify the OMD algorithm through a simple technique that
we call stabilization. We give essentially the same abstract regret bound for
OMD with stabilization and for DA by modifying the classical OMD convergence
analysis in a careful and modular way that allows for straightforward and
flexible proofs. Simple corollaries of these bounds show that OMD with
stabilization and DA enjoy the same performance guarantees in many applications
-- even under dynamic learning rates. We also shed light on the similarities
between OMD and DA and show simple conditions under which stabilized-OMD and DA
generate the same iterates.
