  The performance of online convex optimization algorithms in a dynamic
environment is often expressed in terms of the dynamic regret, which measures
the decision maker's performance against a sequence of time-varying
comparators. In the analysis of the dynamic regret, prior works often assume
Lipschitz continuity or uniform smoothness of the cost functions. However,
there are many important cost functions in practice that do not satisfy these
conditions. In such cases, prior analyses are not applicable and fail to
guarantee the optimization performance. In this letter, we show that it is
possible to bound the dynamic regret, even when neither Lipschitz continuity
nor uniform smoothness is present. We adopt the notion of relative smoothness
with respect to some user-defined regularization function, which is a much
milder requirement on the cost functions. We first show that under relative
smoothness, the dynamic regret has an upper bound based on the path length and
functional variation. We then show that with an additional condition of
relatively strong convexity, the dynamic regret can be bounded by the path
length and gradient variation. These regret bounds provide performance
guarantees to a wide variety of online optimization problems that arise in
different application domains. Finally, we present numerical experiments that
demonstrate the advantage of adopting a regularization function under which the
cost functions are relatively smooth.
