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

• URL: http://arxiv.org/abs/2503.23600v2
• Date: Mon, 14 Apr 2025 15:12:29 GMT
• Title: Online Convex Optimization and Integral Quadratic Constraints: A new approach to regret analysis
• Authors: Fabian Jakob, Andrea Iannelli,
• Abstract summary: We analyze dynamic regret of first-order constrained online convex optimization algorithms for strongly convex and Lipschitz-smooth objectives.<n>We derive a semi-definite program which, when feasible, provides a regret guarantee for the online algorithm.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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