# Policy Gradient Method for LQG Control via Input-Output-History Representation: Convergence to O(ε)-Stationary Points

arXiv: https://arxiv.org/abs/2510.19141 · HTML: https://mlsanity.com/paper/2510.19141/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in RL for games and combinatorial optimization, one of 15 directions in Deep reinforcement learning methods.
- Builds on 15 of its 26 references that are in the map.
- First posted on arXiv 22 Oct 2025.

## Placed in

- Deep reinforcement learning methods / [RL for games and combinatorial optimization](https://mlsanity.com/deeprl/rl-for-games-and-combinatorial-optimization/)

## Builds on (most cited first, 15 of 15)

- [Global Convergence of Policy Gradient Methods for the Linear Quadratic Regulator](https://mlsanity.com/paper/1801.05039/) (2018)
- [Convergence and sample complexity of gradient methods for the model-free linear quadratic regulator problem](https://mlsanity.com/paper/1912.11899/) (2019)
- [LQR through the Lens of First Order Methods: Discrete-time Case](https://mlsanity.com/paper/1907.08921/) (2019)
- [Distributed Reinforcement Learning for Decentralized Linear Quadratic Control: A Derivative-Free Policy Optimization Approach](https://mlsanity.com/paper/1912.09135/) (2019)
- [Analysis of the Optimization Landscape of Linear Quadratic Gaussian (LQG) Control](https://mlsanity.com/paper/2102.04393/) (2021)
- [Optimizing Static Linear Feedback: Gradient Method](https://mlsanity.com/paper/2004.09875/) (2020)
- [Global Convergence of Policy Gradient for Sequential Zero-Sum Linear Quadratic Dynamic Games](https://mlsanity.com/paper/1911.04672/) (2019)
- [Sample Complexity of Linear Quadratic Gaussian (LQG) Control for Output Feedback Systems](https://mlsanity.com/paper/2011.09929/) (2020)
- [On the Optimization Landscape of Dynamic Output Feedback Linear Quadratic Control](https://mlsanity.com/paper/2201.09598/) (2022)
- [On Topological Properties of the Set of Stabilizing Feedback Gains](https://mlsanity.com/paper/1904.08451/) (2019)
- [Finite-Data Performance Guarantees for the Output-Feedback Control of an Unknown System](https://mlsanity.com/paper/1803.09186/) (2018)
- [Optimization Landscape of Policy Gradient Methods for Discrete-time Static Output Feedback](https://mlsanity.com/paper/2310.19022/) (2023)
- [Structured Output Feedback Control for Linear Quadratic Regulator Using Policy Gradient Method](https://mlsanity.com/paper/2203.00289/) (2022)
- [Behavioral Feedback for Optimal LQG Control](https://mlsanity.com/paper/2204.00702/) (2022)
- [Policy Gradient Methods for Designing Dynamic Output Feedback Controllers](https://mlsanity.com/paper/2210.09735/) (2022)

## Built on by (most cited first, 1 of 1)

- [Model-Free Output Feedback Stabilization via Policy Gradient Methods](https://mlsanity.com/paper/2601.19284/) (2026)

## Abstract (arXiv)

We study the policy gradient method (PGM) for the linear quadratic Gaussian (LQG) dynamic output-feedback control problem using an input-output-history (IOH) representation of the closed-loop system. First, we show that any dynamic output-feedback controller is equivalent to a static partial-state feedback gain for a new system representation characterized by a finite-length IOH. Leveraging this equivalence, we reformulate the search for an optimal dynamic output feedback controller as an optimization problem over the corresponding partial-state feedback gain. Next, we introduce a relaxed version of the IOH-based LQG problem by incorporating a small process noise with covariance ε I into the new system to ensure coerciveness, a key condition for establishing gradient-based convergence guarantees. Consequently, we show that a vanilla PGM for the relaxed problem converges to an O(ε)-stationary point, i.e., overlineK satisfying |nabla J(overlineK)|F ≤ O(ε), where J denotes the original LQG cost. Numerical experiments empirically indicate convergence to the vicinity of the globally optimal LQG controller.