Online Optimization with Memory and Competitive Control

TL;DR

This paper introduces competitive algorithms for online optimization problems with memory, extending smoothed convex optimization, and establishes a link to online control with adversarial disturbances, resulting in new control policies.

Contribution

It proposes the Optimistic Regularized Online Balanced Descent algorithm with a dimension-free competitive ratio and connects online optimization with control theory to develop new control policies.

Findings

Achieves a constant, dimension-free competitive ratio.

Establishes a connection between online optimization with memory and control with disturbances.

Develops a new constant-competitive policy for online control problems.

Abstract

This paper presents competitive algorithms for a novel class of online optimization problems with memory. We consider a setting where the learner seeks to minimize the sum of a hitting cost and a switching cost that depends on the previous $p$ decisions. This setting generalizes Smoothed Online Convex Optimization. The proposed approach, Optimistic Regularized Online Balanced Descent, achieves a constant, dimension-free competitive ratio. Further, we show a connection between online optimization with memory and online control with adversarial disturbances. This connection, in turn, leads to a new constant-competitive policy for a rich class of online control problems.