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

arXiv: https://arxiv.org/abs/2310.02930 · HTML: https://mlsanity.com/paper/2310.02930/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Oct 2023.

## Built on by (most cited first, 4 of 4)

- [Remarks on the Polyak-Lojasiewicz inequality and the convergence of gradient systems](https://mlsanity.com/paper/2503.23641/) (2025)
- [On the Robustness of Derivative-free Methods for Linear Quadratic Regulator](https://mlsanity.com/paper/2506.12596/) (2025)
- [A Fully Data-Driven Value Iteration for Stochastic LQR: Convergence, Robustness and Stability](https://mlsanity.com/paper/2505.02970/) (2025)
- [Some remarks on gradient dominance and LQR policy optimization](https://mlsanity.com/paper/2507.10452/) (2025)

## Abstract (arXiv)

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.