Remarks on input to state stability of perturbed gradient flows, motivated by model-free feedback control learning

TL;DR

This paper analyzes the stability of perturbed gradient flows in model-free control, focusing on input-to-state stability and the effects of gradient estimation errors in reinforcement learning.

Contribution

It provides a new stability analysis framework for noisy gradient systems using input-to-state stability concepts, including a review for systems on open subsets of Euclidean space.

Findings

Input-to-state stability applies to gradient-based control systems.

Gradient estimation errors impact the stability of steepest descent algorithms.

The framework aids in understanding robustness in model-free reinforcement learning.

Abstract

Recent work on data-driven control and reinforcement learning has renewed interest in a relative old field in control theory: model-free optimal control approaches which work directly with a cost function and do not rely upon perfect knowledge of a system model. Instead, an "oracle" returns an estimate of the cost associated to, for example, a proposed linear feedback law to solve a linear-quadratic regulator problem. This estimate, and an estimate of the gradient of the cost, might be obtained by performing experiments on the physical system being controlled. This motivates in turn the analysis of steepest descent algorithms and their associated gradient differential equations. This note studies the effect of errors in the estimation of the gradient, framed in the language of input to state stability, where the input represents a perturbation from the true gradient. Since one needs to study systems evolving on proper open subsets of Euclidean space, a self-contained review of input to state stability definitions and theorems for systems that evolve on such sets is included. The results are then applied to the study of noisy gradient systems, as well as the associated steepest descent algorithms.