A Linear Parameter-Varying Framework for the Analysis of Time-Varying Optimization Algorithms

TL;DR

A framework to analyze iterative first-order optimization algorithms for time-varying convex optimization by leveraging the approach of analyzing algorithms as uncertain control interconnections with integral quadratic constraints (IQCs) and generalizing that framework to the time-varying case is proposed.

Abstract

In this paper we propose a framework to analyze iterative first-order optimization algorithms for time-varying convex optimization. We assume that the temporal variability is caused by a time-varying parameter entering the objective, which can be measured at the time of decision but whose future values are unknown. We consider the case of strongly convex objective functions with Lipschitz continuous gradients under a convex constraint set. We model the algorithms as discrete-time linear parameter varying (LPV) systems in feedback with monotone operators such as the time-varying gradient. We leverage the approach of analyzing algorithms as uncertain control interconnections with integral quadratic constraints (IQCs) and generalize that framework to the time-varying case. We propose novel IQCs that are capable of capturing the behavior of time-varying nonlinearities and leverage techniques from the LPV literature to establish novel bounds on the tracking error. Quantitative bounds can be computed by solving a semi-definite program and can be interpreted as an input-to-state stability result with respect to a disturbance signal which increases with the temporal variability of the problem. As a departure from results in this research area, our bounds introduce a dependence on different additional measures of temporal variations, such as the function value and gradient rate of change. We exemplify our main results with numerical experiments that showcase how our analysis framework is able to capture convergence rates of different first-order algorithms for time-varying optimization through the choice of IQC and rate bounds.

Figures (11)

• Figure 1: The first-order algorithm \ref{['eq:algorithm_lpv']} as a feedback interconnection of an LPV system and an oracle $\varphi$, which contains (multiple) evaluations of the gradient.
• Figure 1: Illustration of a two-step method.
• Figure 1: Illustration of the difference between the pointwise and variational IQC filters.
• Figure 1: Augmented algorithm. The influence of the gradient is replaced by an IQC.
• Figure 1: Influence of rate bounds $\overline{\nu}$ and number of algorithm evaluations, for GD.

Theorems & Definitions (23)

• Example 3.1: Accelerated gradient method
• Example 3.2: Two-step gradient descent
• Lemma 3.3
• Proposition 3.4
• Proof 1
• Remark 3.5
• Remark 3.6
• Proposition 4.1: Sector IQC
• Proposition 4.2: Variational IQC
• Remark 4.3