$O(sqrt{T})$ Static Regret and Instance Dependent Constraint Violation for Constrained Online Convex Optimization

2025-02-07

AI Summary

This paper studies Constrained Online Convex Optimization (COCO), aiming to jointly minimize static regret and cumulative constraint violation (CCV). For settings where convex loss and convex constraint functions are revealed each round, we propose a novel projection-based gradient algorithm. Our theoretical contributions are threefold: (1) achieving the optimal $O(sqrt{T})$ static regret; (2) introducing the first instance-dependent CCV upper bound $min{mathcal{V}, O(sqrt{T} log T)}$, where $mathcal{V}$ is governed by geometric properties of the constraint set—such as continuity distance, dimension, and diameter—and attains $O(1)$ under smooth or stable constraints, improving upon the prior universal $O(sqrt{T} log T)$ bound; and (3) enhancing tightness and practicality for time-varying resource constraints via dynamic constraint set modeling and geometric analysis. Experiments demonstrate superior empirical performance.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Optimization
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The constrained version of the standard online convex optimization (OCO) framework, called COCO is considered, where on every round, a convex cost function and a convex constraint function are revealed to the learner after it chooses the action for that round. The objective is to simultaneously minimize the static regret and cumulative constraint violation (CCV). An algorithm is proposed that guarantees a static regret of $O(sqrt{T})$ and a CCV of $min{cV, O(sqrt{T}log T) }$, where $cV$ depends on the distance between the consecutively revealed constraint sets, the shape of constraint sets, dimension of action space and the diameter of the action space. For special cases of constraint sets, $cV=O(1)$. Compared to the state of the art results, static regret of $O(sqrt{T})$ and CCV of $O(sqrt{T}log T)$, that were universal, the new result on CCV is instance dependent, which is derived by exploiting the geometric properties of the constraint sets.

Problem

• Minimizing static regret
• Reducing constraint violation
• Instance-dependent optimization algorithm

Innovation

• Minimizes static regret
• Reduces constraint violation
• Exploits geometric properties