# Perturbed Gradient Descent Algorithms are Small-Disturbance Input-to-State Stable

arXiv: https://arxiv.org/abs/2507.02131 · HTML: https://mlsanity.com/paper/2507.02131/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jul 2025.

## Built on by (most cited first, 3 of 3)

- [Learning to optimize with guarantees: a complete characterization of linearly convergent algorithms](https://mlsanity.com/paper/2508.00775/) (2025)
- [A Stochastic Gradient Descent Approach to Design Policy Gradient Methods for LQR](https://mlsanity.com/paper/2602.18933/) (2026)
- [Simplicity Suffices for Parameter Noise Injection in Stochastic Gradient Descent](https://mlsanity.com/paper/2606.12054/) (2026)

## Abstract (arXiv)

This article investigates the robustness of gradient descent algorithms under perturbations. The concept of small-disturbance input-to-state stability (ISS) for discrete-time nonlinear dynamical systems is introduced, along with its Lyapunov characterization. The conventional linear Polyak-Lojasiewicz (PL) condition is then extended to a nonlinear version, and it is shown that the gradient descent algorithm is small-disturbance ISS provided the objective function satisfies the generalized nonlinear PL condition. This small-disturbance ISS property guarantees that the gradient descent algorithm converges to a small neighborhood of the optimum under sufficiently small perturbations. As a direct application of the developed framework, we demonstrate that the LQR cost satisfies the generalized nonlinear PL condition, thereby establishing that the policy gradient algorithm for LQR is small-disturbance ISS. Additionally, other popular policy gradient algorithms, including natural policy gradient and Gauss-Newton method, are also proven to be small-disturbance ISS.