Perturbed Gradient Descent Algorithms are Small-Disturbance Input-to-State Stable
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916824101158912 |
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| author | Cui, Leilei Jiang, Zhong-Ping Sontag, Eduardo D. Braatz, Richard D. |
| author_facet | Cui, Leilei Jiang, Zhong-Ping Sontag, Eduardo D. Braatz, Richard D. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02131 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Perturbed Gradient Descent Algorithms are Small-Disturbance Input-to-State Stable Cui, Leilei Jiang, Zhong-Ping Sontag, Eduardo D. Braatz, Richard D. Optimization and Control Systems and Control 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. |
| title | Perturbed Gradient Descent Algorithms are Small-Disturbance Input-to-State Stable |
| topic | Optimization and Control Systems and Control |
| url | https://arxiv.org/abs/2507.02131 |