# Learning Stabilizing Policies via an Unstable Subspace Representation

arXiv: https://arxiv.org/abs/2505.01348 · HTML: https://mlsanity.com/paper/2505.01348/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 40 references that are in the map.
- First posted on arXiv 2 May 2025.

## Builds on (most cited first, 19 of 19)

- [Global Convergence of Policy Gradient Methods for the Linear Quadratic Regulator](https://mlsanity.com/paper/1801.05039/) (2018)
- [Near optimal finite time identification of arbitrary linear dynamical systems](https://mlsanity.com/paper/1812.01251/) (2018)
- [Derivative-Free Methods for Policy Optimization: Guarantees for Linear Quadratic Systems](https://mlsanity.com/paper/1812.08305/) (2018)
- [Convergence and sample complexity of gradient methods for the model-free linear quadratic regulator problem](https://mlsanity.com/paper/1912.11899/) (2019)
- [Black-Box Control for Linear Dynamical Systems](https://mlsanity.com/paper/2007.06650/) (2020)
- [Stabilizing Dynamical Systems via Policy Gradient Methods](https://mlsanity.com/paper/2110.06418/) (2021)
- [Model-free Learning with Heterogeneous Dynamical Systems: A Federated LQR Approach](https://mlsanity.com/paper/2308.11743/) (2023)
- [Meta-Learning Linear Quadratic Regulators: A Policy Gradient MAML Approach for Model-free LQR](https://mlsanity.com/paper/2401.14534/) (2024)
- [Sample-Efficient Linear Representation Learning from Non-IID Non-Isotropic Data](https://mlsanity.com/paper/2308.04428/) (2023)
- [Online Policy Gradient for Model Free Learning of Linear Quadratic Regulators with √T Regret](https://mlsanity.com/paper/2102.12608/) (2021)
- [Learning to Control Linear Systems can be Hard](https://mlsanity.com/paper/2205.14035/) (2022)
- [On the Sample Complexity of Stabilizing LTI Systems on a Single Trajectory](https://mlsanity.com/paper/2202.07187/) (2022)
- [Convergence and Sample Complexity of Policy Gradient Methods for Stabilizing Linear Systems](https://mlsanity.com/paper/2205.14335/) (2022)
- [Coreset-Based Task Selection for Sample-Efficient Meta-Reinforcement Learning](https://mlsanity.com/paper/2502.02332/) (2025)
- [Asynchronous Heterogeneous Linear Quadratic Regulator Design](https://mlsanity.com/paper/2404.09061/) (2024)
- [Regret Analysis of Multi-task Representation Learning for Linear-Quadratic Adaptive Control](https://mlsanity.com/paper/2407.05781/) (2024)
- [Learning to Stabilize Unknown LTI Systems on a Single Trajectory under Stochastic Noise](https://mlsanity.com/paper/2406.00234/) (2024)
- [System stabilization with policy optimization on unstable latent manifolds](https://mlsanity.com/paper/2407.06418/) (2024)
- [On the Hardness of Learning to Stabilize Linear Systems](https://mlsanity.com/paper/2311.11151/) (2023)

## Built on by (most cited first, 3 of 3)

- [Adversarially Robust Multitask Adaptive Control](https://mlsanity.com/paper/2511.05444/) (2025)
- [Model-Free Output Feedback Stabilization via Policy Gradient Methods](https://mlsanity.com/paper/2601.19284/) (2026)
- [Symplectic Inductive Bias for Data-Driven Target Reachability in Hamiltonian Systems](https://mlsanity.com/paper/2604.17213/) (2026)

## Abstract (arXiv)

We study the problem of learning to stabilize (LTS) a linear time-invariant (LTI) system. Policy gradient (PG) methods for control assume access to an initial stabilizing policy. However, designing such a policy for an unknown system is one of the most fundamental problems in control, and it may be as hard as learning the optimal policy itself. Existing work on the LTS problem requires large data as it scales quadratically with the ambient dimension. We propose a two-phase approach that first learns the left unstable subspace of the system and then solves a series of discounted linear quadratic regulator (LQR) problems on the learned unstable subspace, targeting to stabilize only the system's unstable dynamics and reduce the effective dimension of the control space. We provide non-asymptotic guarantees for both phases and demonstrate that operating on the unstable subspace reduces sample complexity. In particular, when the number of unstable modes is much smaller than the state dimension, our analysis reveals that LTS on the unstable subspace substantially speeds up the stabilization process. Numerical experiments are provided to support this sample complexity reduction achieved by our approach.