Online Convex Optimization Using Coordinate Descent Algorithms

TL;DR

This paper extends coordinate descent algorithms to online convex optimization with time-varying objectives, providing regret analysis and numerical validation for different update rules.

Contribution

It introduces online coordinate descent algorithms for time-varying functions and analyzes their regret, bridging offline methods with online optimization.

Findings

Regret bounds are established for various coordinate descent rules.

Numerical simulations confirm the theoretical regret performance.

The approach handles both deterministic and stochastic update rules.

Abstract

This paper considers the problem of online optimization where the objective function is time-varying. In particular, we extend coordinate descent type algorithms to the online case, where the objective function varies after a finite number of iterations of the algorithm. Instead of solving the problem exactly at each time step, we only apply a finite number of iterations at each time step. Commonly used notions of regret are used to measure the performance of the online algorithm. Moreover, coordinate descent algorithms with different updating rules are considered, including both deterministic and stochastic rules that are developed in the literature of classical offline optimization. A thorough regret analysis is given for each case. Finally, numerical simulations are provided to illustrate the theoretical results.