  In online convex optimization (OCO), Lipschitz continuity of the functions is
commonly assumed in order to obtain sublinear regret. Moreover, many algorithms
have only logarithmic regret when these functions are also strongly convex.
Recently, researchers from convex optimization proposed the notions of
"relative Lipschitz continuity" and "relative strong convexity". Both of the
notions are generalizations of their classical counterparts. It has been shown
that subgradient methods in the relative setting have performance analogous to
their performance in the classical setting.
  In this work, we consider OCO for relative Lipschitz and relative strongly
convex functions. We extend the known regret bounds for classical OCO
algorithms to the relative setting. Specifically, we show regret bounds for the
follow the regularized leader algorithms and a variant of online mirror
descent. Due to the generality of these methods, these results yield regret
bounds for a wide variety of OCO algorithms. Furthermore, we further extend the
results to algorithms with extra regularization such as regularized dual
averaging.
