{"ID":2899374,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.02131","arxiv_id":"2507.02131","title":"Perturbed Gradient Descent Algorithms are Small-Disturbance Input-to-State Stable","abstract":"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.","short_abstract":"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...","url_abs":"https://arxiv.org/abs/2507.02131","url_pdf":"https://arxiv.org/pdf/2507.02131v1","authors":"[\"Leilei Cui\",\"Zhong-Ping Jiang\",\"Eduardo D. Sontag\",\"Richard D. Braatz\"]","published":"2025-07-02T20:31:28Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"eess.SY\"]","methods":"[]","has_code":false}