Polyak–Łojasiewicz Inequality and Backtracking Line‐search in Iterative Feedback Tuning

ABSTRACT

This paper explores the Polyak–Łojasiewicz inequality and presents an associated convergence analysis of the gradient method with the backtracking line‐search in the iterative feedback tuning (IFT). In the conventional IFT, because stepsize design requires the information of a target unknown model, theoretical convergence guarantee is unavailable in practical applications, and the stepsize design has been developed in terms of numerical experiments in existing studies. Even if the aforementioned stepsize, depending on a part of the model information, can be ideally available, convergence rate has hardly been studied. Under some assumptions, this paper derives a PŁ inequality, and the linear convergence is established by exploiting the gradient descent condition using the backtracking line‐search, which leads to an automated stepsize search mechanism with the theoretical convergence guarantee.