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Online Convex Optimization Using Coordinate Descent Algorithms

Yankai Lin, Iman Shames, Dragan Nešić

TL;DR

This work extends online convex optimization to time-varying objectives by adapting coordinate descent methods to the online setting. It analyzes both random and deterministic coordinate updates, deriving regret bounds for static and dynamic regimes under convex and strongly convex assumptions, and shows that sublinear regret rates comparable to full-gradient online methods are attainable. Key contributions include rigorous regret analyses for random, cyclic, and Gauss-Southwell coordinate rules, the use of the doubling trick for adaptive stepsizes, and extensions to multi-update per step in the strongly convex regime. Numerical simulations validate the theoretical bounds and highlight practical performance differences among coordinate schemes, with Gauss-Southwell frequently performing best among coordinate updates. The results demonstrate the viability of online coordinate descent as a scalable alternative for large-scale, time-varying optimization problems.

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.

Online Convex Optimization Using Coordinate Descent Algorithms

TL;DR

This work extends online convex optimization to time-varying objectives by adapting coordinate descent methods to the online setting. It analyzes both random and deterministic coordinate updates, deriving regret bounds for static and dynamic regimes under convex and strongly convex assumptions, and shows that sublinear regret rates comparable to full-gradient online methods are attainable. Key contributions include rigorous regret analyses for random, cyclic, and Gauss-Southwell coordinate rules, the use of the doubling trick for adaptive stepsizes, and extensions to multi-update per step in the strongly convex regime. Numerical simulations validate the theoretical bounds and highlight practical performance differences among coordinate schemes, with Gauss-Southwell frequently performing best among coordinate updates. The results demonstrate the viability of online coordinate descent as a scalable alternative for large-scale, time-varying optimization problems.

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.
Paper Structure (15 sections, 8 theorems, 87 equations, 3 figures, 1 table, 5 algorithms)

This paper contains 15 sections, 8 theorems, 87 equations, 3 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Online coordinate descent algorithms
  4. Regret bounds for online coordinate descent algorithms with random coordinate selection rules
  5. Static regret for convex functions
  6. Static regret for strongly convex functions
  7. Dynamic regret for convex functions
  8. Dynamic regret for strongly convex functions
  9. Regret bounds for online coordinate descent algorithms with deterministic coordinate selection rules
  10. Static regret for convex functions
  11. Static regret for strongly convex functions
  12. Dynamic regret for convex functions
  13. Dynamic regret for strongly convex functions
  14. Numerical simulations
  15. Summary

Key Result

Lemma 1

Bert1 Let $\Theta$ be a non-empty closed convex set. We have $|\Pi_{\Theta}(x)-\Pi_{\Theta}(y)|\leq|x-y|$, for any $x,y\in\mathbb{R}^n$.

Figures (3)

  • Figure 1: Plots of the dynamic regrets $R_T^d$ and their time-averages $R_T^d/T$.
  • Figure 2: Plots of the dynamic regrets $R_T^d$ and their time-averages $R_T^d/T$ with slow variation.
  • Figure 3: Plots of the dynamic regrets $R_T^d$ and their time-averages $R_T^d/T$ in the non-strongly convex case.

Theorems & Definitions (18)

  • Remark 1
  • Lemma 1
  • Remark 2
  • Remark 3
  • Definition 1: Doubling trick scheme
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Proposition 1
  • ...and 8 more