Analysis of Gradient Descent with Varying Step Sizes using Integral Quadratic Constraints

In this paper, we present an analysis of the convergence rate of gradient descent with a varying step size, on strongly convex functions. We assume that a line search has been carried out and produces a step size that varies in a known interval. The algorithm is then handled as a linear, parameter-varying (LPV) system. Building on prior work that uses Integral Quadratic Constraints (IQCs) to analyze optimization algorithms, we construct a linear matrix inequality (LMI) condition to numerically obtain convergence rates. For the LPV system, this condition is solved by a gridding approach on the step size interval. Our results indicate that the algorithm converges in a restricted set within the step size interval. Further, when this interval reduces to a point, i.e., when a constant step size is used, the algorithm recovers the gradient rate corresponding to constant step sizes.Comment: Submitted to ACC 2023. 6 pages, 4 figure