  The framework of Integral Quadratic Constraints (IQCs) is used to perform an
analysis of gradient descent with varying step sizes. Two performance metrics
are considered: convergence rate and noise amplification. We assume that the
step size is produced from a line search and varies in a known interval.
Modeling the algorithm as a linear, parameter-varying (LPV) system, we
construct a parameterized linear matrix inequality (LMI) condition that
certifies algorithm performance, which is solved using a result for polytopic
LPV systems. Our results provide convergence rate guarantees when the step size
lies within a restricted interval. Moreover, we recover existing rate bounds
when this interval reduces to a single point, i.e. a constant step size.
Finally, we note that the convergence rate depends only on the condition number
of the problem. In contrast, the noise amplification performance depends on the
individual values of the strong convexity and smoothness parameters, and varies
inversely with them for a fixed condition number.
