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Universal Dynamic Regret and Constraint Violation Bounds for Constrained Online Convex Optimization

Subhamon Supantha, Abhishek Sinha

TL;DR

This work considers a generalization of the celebrated Online Convex Optimization framework with online adversarial constraints and presents two algorithms having simple modular structures that yield universal dynamic regret and cumulative constraint violation bounds, improving upon the state-of-the-art results.

Abstract

We consider a generalization of the celebrated Online Convex Optimization (OCO) framework with online adversarial constraints. We present two algorithms having simple modular structures that yield universal dynamic regret and cumulative constraint violation bounds, improving upon the state-of-the-art results. Our results hold in the most general case when both the cost and constraint functions are chosen arbitrarily by an adversary, and the constraint functions need not contain any common feasible point. The results are established by reducing the constrained learning problem to an instance of the standard OCO problem with specially constructed surrogate cost functions.

Universal Dynamic Regret and Constraint Violation Bounds for Constrained Online Convex Optimization

TL;DR

This work considers a generalization of the celebrated Online Convex Optimization framework with online adversarial constraints and presents two algorithms having simple modular structures that yield universal dynamic regret and cumulative constraint violation bounds, improving upon the state-of-the-art results.

Abstract

We consider a generalization of the celebrated Online Convex Optimization (OCO) framework with online adversarial constraints. We present two algorithms having simple modular structures that yield universal dynamic regret and cumulative constraint violation bounds, improving upon the state-of-the-art results. Our results hold in the most general case when both the cost and constraint functions are chosen arbitrarily by an adversary, and the constraint functions need not contain any common feasible point. The results are established by reducing the constrained learning problem to an instance of the standard OCO problem with specially constructed surrogate cost functions.

Paper Structure

This paper contains 27 sections, 8 theorems, 57 equations, 2 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Works
  4. COCO with Static Regret Guarantees:
  5. COCO with Dynamic Regret Guarantees:
  6. Convex Body Chasing:
  7. Problem Statement and Performance Metrics
  8. Comparison between the Universal and Worst-Case Dynamic Regret Metrics:
  9. COCO Algorithm with Full Feedback
  10. Bounding CCV:
  11. Bounding Regret:
  12. Computational Aspects:
  13. COCO Algorithm with First-Order Feedback
  14. Preliminaries: Universal Dynamic Regret with Unbounded Lipschitz Constants
  15. Tightening the Universal Dynamic Regret Bound with Multiple Experts
  16. ...and 12 more sections

Key Result

lemma 1

For any feasible comparator sequence $u_{1:T}$ and any decision sequence $x_{1:T}$, we have $\textsc{UD-Regret}(f_{1:T};u_{1:T}) \leq \textsc{UD-Regret}(\tilde{f}_{1:T};u_{1:T}),$i.e.,

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (24)

  • remark 1
  • definition 1: Comparator Sequence
  • definition 2: Path length
  • definition 3: Feasible Set
  • definition 4: Feasible Comparators
  • definition 5: Cumulative Constraint Violation (CCV)
  • definition 6: Universal Dynamic Regret
  • remark 2
  • remark 3
  • definition 7: Worst-case Dynamic Regret
  • ...and 14 more