Learning Stabilizing Policies via an Unstable Subspace Representation (2505.01348v1)

Abstract: 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.

Insight into Learning Stabilizing Policies via an Unstable Subspace Representation

The paper "Learning Stabilizing Policies via an Unstable Subspace Representation" presents a method that addresses the challenge of stabilizing Linear Time-Invariant (LTI) systems, particularly those with a large state dimension but a relatively small number of unstable modes. Traditional stabilization methods for LTI systems often require the identification of stabilizing policies, which is a challenging task when the system is unknown. The paper proposes a novel approach that efficiently leverages the unstable subspace to simplify and expedite this stabilization process.

Core Contributions

The primary contributions of this work can be summarized as follows:

1. Two-Phase Stabilization Approach: The authors designed a two-phase method to tackle the learning to stabilize (LTS) problem. The first phase involves learning the left unstable subspace of the system, which has reduced dimensionality. The second phase employs a series of discounted Linear Quadratic Regulator (LQR) problems focused on this learned subspace, significantly reducing sample complexity and dimensionality compared to traditional methods that operate on the full system representation.
2. Non-Asymptotic Guarantees: The paper provides non-asymptotic guarantees for both phases of the approach, ensuring that these methods are effective in practice. Specifically, the reduction in sample complexity is pronounced when the number of unstable modes is substantially smaller than the overall state dimension.
3. Numerical Validation: Extensive numerical experiments validate the theoretical results, demonstrating that solving LQR problems over the unstable subspace is more sample efficient than existing methods that do not take advantage of subspace reduction.

Implications and Future Directions

The implications of this research are promising for the development of more efficient stabilization techniques in control theory and reinforcement learning applications. By focusing on the unstable subspace, this approach circumvents the prohibitive costs associated with handling high-dimensional systems directly. Practically, this means that control systems can be stabilized with fewer samples and less computational overhead.

Theoretically, this work opens avenues for exploring the stabilization of complex systems by zeroing in on critical subspaces, especially in fields where data acquisition costs are high, such as robotics and autonomous systems. Moreover, as mentioned, the method accommodates non-diagonalizable systems, which significantly broadens its applicability.

In terms of future directions, the authors suggest adapting these techniques for multi-system setups where multiple systems share similar unstable subspaces, thereby enabling efficient learning across tasks. Another potential area for exploration could be the online adaptation of subspace learning, which could further improve model adaptability.

Conclusion

The approach and insights presented in "Learning Stabilizing Policies via an Unstable Subspace Representation" offer a compelling method for efficiently stabilizing LTI systems, especially when the size and complexity of the system present challenges in traditional control settings. The reduction in sample complexity while maintaining stability highlights the utility of focusing on unstable subspaces, inspiring future research to build on these foundations and extend them to broader and more application-specific contexts.