  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.
