Online Convex Optimization and Integral Quadratic Constraints: An automated approach to regret analysis
This work provides a more flexible regret analysis framework for online convex optimization, which is incremental as it extends existing IQC methods to time-varying settings.
The paper tackles the problem of analyzing dynamic regret for constrained online convex optimization algorithms by proposing a novel approach using Integral Quadratic Constraints (IQCs) to derive regret guarantees without requiring gradient boundedness or bounded feasible sets, resulting in a general analysis applicable to a wide range of first-order algorithms.
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.