Online Convex Optimization with Long Term Constraints for Predictable Sequences

TL;DR

This paper introduces a new online convex optimization algorithm that leverages predictability in cost sequences with long term constraints, achieving lower regret and constraint violation rates than non-predictive methods.

Contribution

It proposes a novel algorithm for online convex optimization with long term constraints that exploits sequence predictability to improve performance.

Findings

Achieves lower regret with predictive information.

Reduces constraint violation compared to non-predictive algorithms.

Demonstrates effectiveness in predictable sequence settings.

Abstract

In this paper, 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 {\it OCO with long term constraints}. Long term constraints are introduced typically as an alternative to reduce the complexity of the 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 $T$ 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 paper, we study the setting where the sequences are predictable. We present a novel online optimization algorithm for online optimization with long term constraints that can leverage such predictability. 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.