  Dual-based proportional-integral-derivative (PID) controllers are often
employed in practice to solve online allocation problems with global
constraints, such as budget pacing in online advertising. However, controllers
are used in a heuristic fashion and come with no provable guarantees on their
performance. This paper provides the first regret bounds on the performance of
dual-based PID controllers for online allocation problems. We do so by first
establishing a fundamental connection between dual-based PID controllers and a
new first-order algorithm for online convex optimization called
\emph{Convolutional Mirror Descent} (CMD), which updates iterates based on a
weighted moving average of past gradients. CMD recovers, in a special case,
online mirror descent with momentum and optimistic mirror descent. We establish
sufficient conditions under which CMD attains low regret for general online
convex optimization problems with adversarial inputs. We leverage this new
result to give the first regret bound for dual-based PID controllers for online
allocation problems. As a byproduct of our proofs, we provide the first regret
bound for CMD for non-smooth convex optimization, which might be of independent
interest.
