Small-Disturbance Input-to-State Stability of Perturbed Gradient Flows: Applications to LQR Problem

TL;DR

This paper demonstrates that perturbed gradient flows in nonlinear programming, including LQR problems, are robust to small disturbances, ensuring convergence to near-optimal solutions despite inaccuracies or errors.

Contribution

It establishes small-disturbance input-to-state stability for various gradient flows in nonlinear optimization, including LQR, under perturbations from gradient estimation errors.

Findings

Perturbed gradient flows are ISS under certain conditions.

Standard, natural, and Newton gradient flows exhibit small-disturbance ISS.

Trajectories converge to a neighborhood of the optimum despite perturbations.

Abstract

This paper studies the effect of perturbations on the gradient flow of a general nonlinear programming problem, where the perturbation may arise from inaccurate gradient estimation in the setting of data-driven optimization. Under suitable conditions on the objective function, the perturbed gradient flow is shown to be small-disturbance input-to-state stable (ISS), which implies that, in the presence of a small-enough perturbation, the trajectories of the perturbed gradient flow must eventually enter a small neighborhood of the optimum. This work was motivated by the question of robustness of direct methods for the linear quadratic regulator problem, and specifically the analysis of the effect of perturbations caused by gradient estimation or round-off errors in policy optimization. We show small-disturbance ISS for three of the most common optimization algorithms: standard gradient flow, natural gradient flow, and Newton gradient flow.