# On the Gradient Domination of the LQG Problem

arXiv: https://arxiv.org/abs/2507.09026 · HTML: https://mlsanity.com/paper/2507.09026/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL for games and combinatorial optimization, one of 15 directions in Deep reinforcement learning methods.
- Builds on 4 of its 30 references that are in the map.
- First posted on arXiv 11 Jul 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, 4 of 4)

- [Model-free Learning with Heterogeneous Dynamical Systems: A Federated LQR Approach](https://mlsanity.com/paper/2308.11743/) (2023)
- [Robot Fleet Learning via Policy Merging](https://mlsanity.com/paper/2310.01362/) (2023)
- [Coreset-Based Task Selection for Sample-Efficient Meta-Reinforcement Learning](https://mlsanity.com/paper/2502.02332/) (2025)
- [Learning Stabilizing Policies via an Unstable Subspace Representation](https://mlsanity.com/paper/2505.01348/) (2025)

## Abstract (arXiv)

We consider solutions to the linear quadratic Gaussian (LQG) regulator problem via policy gradient (PG) methods. Although PG methods have demonstrated strong theoretical guarantees in solving the linear quadratic regulator (LQR) problem, despite its nonconvex landscape, their theoretical understanding in the LQG setting remains limited. Notably, the LQG problem lacks gradient dominance in the classical parameterization, i.e., with a dynamic controller, which hinders global convergence guarantees. In this work, we study PG for the LQG problem by adopting an alternative parameterization of the set of stabilizing controllers and employing a lifting argument. We refer to this parameterization as a history representation of the control input as it is parameterized by past input and output data from the previous p time-steps. This representation enables us to establish gradient dominance and approximate smoothness for the LQG cost. We prove global convergence and per-iteration stability guarantees for policy gradient LQG in model-based and model-free settings. Numerical experiments on an open-loop unstable system are provided to support the global convergence guarantees and to illustrate convergence under different history lengths of the history representation.