Online Convex Optimization With Long-Term Constraints for Predictable Sequences

Abstract

In this letter, we investigate the framework of Online Convex Optimization (OCO) for online learning. OCO offers a very powerful online learning framework for many applications. In this context, we study a specific framework of OCO called OCO with long-term constraints. Long-term constraints are introduced typically as an alternative to reduce the complexity of projection at every update step in online optimization. While many algorithmic advances have been made towards online optimization with long-term constraints, these algorithms typically assume that the sequence of cost functions over a certain <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> finite steps that determine the cost to the online learner are adversarially generated. In many circumstances, the sequence of cost functions may not be unrelated, and thus predictable from those observed till a point of time. In this letter, we study the setting where the sequences are predictable. We present a novel algorithm for online optimization with long-term constraints that can leverage such predictability for linear cost functions. We show that, with a predictor that can supply the gradient information of the next function in the sequence, our algorithm can achieve an overall regret and constraint violation rate that is strictly less than the rate that is achievable without prediction.