  The information-theoretic analysis by Russo and Van Roy (2014) in combination
with minimax duality has proved a powerful tool for the analysis of online
learning algorithms in full and partial information settings. In most
applications there is a tantalising similarity to the classical analysis based
on mirror descent. We make a formal connection, showing that the
information-theoretic bounds in most applications can be derived from existing
techniques for online convex optimisation. Besides this, for $k$-armed
adversarial bandits we provide an efficient algorithm with regret that matches
the best information-theoretic upper bound and improve best known regret
guarantees for online linear optimisation on $\ell_p$-balls and bandits with
graph feedback.
