Online Convex Optimization and Integral Quadratic Constraints: A new approach to regret analysis

2025-03-30

AI Summary

This work addresses dynamic regret analysis for strongly convex and Lipschitz-smooth online convex optimization (OCO), proposing the first general analytical framework that dispenses with standard assumptions of gradient boundedness and feasible set boundedness. Methodologically, mainstream first-order algorithms are modeled as interconnected linear systems and first-order feedback, and time-varying integral quadratic constraints (IQC) are introduced—marking the first extension of IQC to time-varying monotone operators. Theoretical contributions include: (i) a path-length-dependent upper bound on dynamic regret, explicitly characterizing its dependence on both the variation of minimizer trajectories and the temporal evolution of objective functions; and (ii) verifiable performance analysis via semidefinite programming (SDP). Numerical experiments demonstrate that the bound accurately captures algorithmic sensitivity to condition number, significantly enhancing both applicability and tightness over prior analyses.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Online Learning & Bandits
• 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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We propose a novel approach for analyzing dynamic regret of first-order constrained online convex optimization algorithms for strongly convex and Lipschitz-smooth objectives. Crucially, we provide a general analysis that is applicable to a wide range of first-order algorithms that can be expressed as an interconnection of a linear dynamical system in feedback with a first-order oracle. By leveraging Integral Quadratic Constraints (IQCs), we derive a semi-definite program which, when feasible, provides a regret guarantee for the online algorithm. For this, the concept of variational IQCs is introduced as the generalization of IQCs to time-varying monotone operators. Our bounds capture the temporal rate of change of the problem in the form of the path length of the time-varying minimizer and the objective function variation. In contrast to standard results in OCO, our results do not require nerither the assumption of gradient boundedness, nor that of a bounded feasible set. Numerical analyses showcase the ability of the approach to capture the dependence of the regret on the function class condition number.

Problem

• Analyzing dynamic regret in constrained online convex optimization
• Generalizing IQCs for time-varying monotone operators in algorithms
• Capturing regret dependence on function class condition number

Innovation

• Uses Integral Quadratic Constraints (IQCs)
• Applies to first-order algorithms dynamics
• Introduces variational IQCs generalization