# Optimization Landscape of Gradient Descent for Discrete-time Static Output Feedback

arXiv: https://arxiv.org/abs/2109.13132 · HTML: https://mlsanity.com/paper/2109.13132/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Sep 2021.

## Built on by (most cited first, 9 of 9)

- [Global Convergence of Two-timescale Actor-Critic for Solving Linear Quadratic Regulator](https://mlsanity.com/paper/2208.08744/) (2022)
- [Global Convergence of Direct Policy Search for State-Feedback H_∞ Robust Control: A Revisit of Nonsmooth Synthesis with Goldstein Subdifferential](https://mlsanity.com/paper/2210.11577/) (2022)
- [Towards a Theoretical Foundation of Policy Optimization for Learning Control Policies](https://mlsanity.com/paper/2210.04810/) (2022)
- [Controlgym: Large-Scale Control Environments for Benchmarking Reinforcement Learning Algorithms](https://mlsanity.com/paper/2311.18736/) (2023)
- [On the Optimization Landscape of Dynamic Output Feedback: A Case Study for Linear Quadratic Regulator](https://mlsanity.com/paper/2209.05042/) (2022)
- [Model-Free Output Feedback Stabilization via Policy Gradient Methods](https://mlsanity.com/paper/2601.19284/) (2026)
- [Accelerated Optimization Landscape of Linear-Quadratic Regulator](https://mlsanity.com/paper/2307.03590/) (2023)
- [Model-Free μ-Synthesis: A Nonsmooth Optimization Perspective](https://mlsanity.com/paper/2402.11654/) (2024)
- [Two-Timescale Optimization Framework for Sparse-Feedback Linear-Quadratic Optimal Control](https://mlsanity.com/paper/2406.11168/) (2024)

## Abstract (arXiv)

In this paper, we analyze the optimization landscape of gradient descent methods for static output feedback (SOF) control of discrete-time linear time-invariant systems with quadratic cost. The SOF setting can be quite common, for example, when there are unmodeled hidden states in the underlying process. We first establish several important properties of the SOF cost function, including coercivity, L-smoothness, and M-Lipschitz continuous Hessian. We then utilize these properties to show that the gradient descent is able to converge to a stationary point at a dimension-free rate. Furthermore, we prove that under some mild conditions, gradient descent converges linearly to a local minimum if the starting point is close to one. These results not only characterize the performance of gradient descent in optimizing the SOF problem, but also shed light on the efficiency of general policy gradient methods in reinforcement learning.