Episodic Linear Quadratic Regulators with Low-rank Transitions
This work addresses the problem of sample-efficient control learning for high-dimensional systems, offering a significant improvement over traditional methods by exploiting low-rank transitions, though it is incremental as it builds on existing LQR frameworks.
The paper tackles the inefficiency of learning Linear Quadratic Regulators (LQR) with unknown dynamics in high-dimensional scenarios, such as when states are high-resolution images, by proposing an algorithm that leverages low-rank structure to achieve a regret bound of order ̃O(m^{3/2} K^{1/2}) and sample complexity dependent only on rank m rather than ambient dimension d.
Linear Quadratic Regulators (LQR) achieve enormous successful real-world applications. Very recently, people have been focusing on efficient learning algorithms for LQRs when their dynamics are unknown. Existing results effectively learn to control the unknown system using number of episodes depending polynomially on the system parameters, including the ambient dimension of the states. These traditional approaches, however, become inefficient in common scenarios, e.g., when the states are high-resolution images. In this paper, we propose an algorithm that utilizes the intrinsic system low-rank structure for efficient learning. For problems of rank-$m$, our algorithm achieves a $K$-episode regret bound of order $\widetilde{O}(m^{3/2} K^{1/2})$. Consequently, the sample complexity of our algorithm only depends on the rank, $m$, rather than the ambient dimension, $d$, which can be orders-of-magnitude larger.