On Dual-Based PI Controllers for Online Allocation Problems

Dual-based proportional-integral (PI) 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 PI controllers for online allocation problems. We do so by first establishing a fundamental connection between dual-based PI controllers and a new first-order algorithm for online convex optimization, which, in a special case, recovers online mirror descent with momentum. We prove the proposed first-order algorithm 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 PI controllers for online allocation problems. As a byproduct of our proofs, we provide the first regret bound for online mirror descent for non-smooth convex optimization, which might be of independent interest.