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Inspired by the work of Tsiamis et al. [1], in this paper we study the statistical hardness of learning to stabilize linear time-invariant systems. Hardness is measured by the number of samples required to achieve a learning task with a given probability. The work in [1] shows that there exist system classes that are hard to learn to stabilize with the core reason being the hardness of identification. Here we present a class of systems that can be easy to identify, thanks to a non-degenerate noise process that excites all modes, but the sample complexity of stabilization still increases exponentially with the system dimension. We tie this result to the hardness of co-stabilizability for this class of systems using ideas from robust control. 

While Markov jump systems (MJSs) are more appropriate than LTI systems in terms of modeling abruptly changing dynamics, MJSs (and other switched systems) may suffer from the model complexity brought by the potentially sheer number of switching modes. Much of the existing work on reducing switched systems focuses on the state space where techniques such as discretization and dimension reduction are performed, yet reducing mode complexity receives few attention. In this work, inspired by clustering techniques from unsupervised learning, we propose a reduction method for MJS such that a mode-reduced MJS can be constructed with guaranteed approximation performance. Furthermore, we show how this reduced MJS can be used in designing controllers for the original MJS to reduce the computation cost while maintaining guaranteed suboptimality. 

While Markov jump systems (MJSs) are more appropriate than LTI systems in terms of modeling abruptly changing dynamics, MJSs (and other switched systems) may suffer from the model complexity brought by the potentially sheer number of switching modes. Much of the existing work on reducing switched systems focuses on the state space where techniques such as discretization and dimension reduction are performed, yet reducing mode complexity receives few attention. In this work, inspired by clustering techniques from unsupervised learning, we propose a reduction method for MJS such that a mode-reduced MJS can be constructed with guaranteed approximation performance. Furthermore, we show how this reduced MJS can be used in designing controllers for the original MJS to reduce the computation cost while maintaining guaranteed suboptimality. Keywords: Markov Jump Systems, System Reduction, Clustering 

While Markov jump systems (MJSs) are more appropriate than LTI systems in terms of modeling abruptly changing dynamics, MJSs (and other switched systems) may suffer from the model complexity brought by the potentially sheer number of switching modes. Much of the existing work on reducing switched systems focuses on the state space where techniques such as discretization and dimension reduction are performed, yet reducing mode complexity receives few attention. In this work, inspired by clustering techniques from unsupervised learning, we propose a reduction method for MJS such that a mode-reduced MJS can be constructed with guaranteed approximation performance. Furthermore, we show how this reduced MJS can be used in designing controllers for the original MJS to reduce the computation cost while maintaining guaranteed suboptimality. 

AutoRegressive eXogenous (ARX) models form one of the most important model classes in control theory, econometrics, and statistics, but they are yet to be understood in terms of their finite sample identification analysis. The technical challenges come from the strong statistical dependency not only between data samples at different time instances but also between elements within each individual sample. In this work, for ARX models with potentially unknown orders, we study how ordinary least squares (OLS) estimator performs in terms of identifying model parameters from data collected from either a single length-T trajectory or N i.i.d. trajectories. Our main results show that as long as the orders of the model are chosen optimistically, i.e., we are learning an over-parameterized model compared to the ground truth ARX, the OLS will converge with the optimal rate O(1/√T) (or O(1/√N)) to the true (low-order) ARX parameters. This occurs without the aid of any regularization, thus is referred to as self-regularization. Our results imply that the oracle knowledge of the true orders and usage of regularizers are not necessary in learning ARX models — over-parameterization is all you need. 

AutoRegressive eXogenous (ARX) models form one of the most important model classes in control theory, econometrics, and statistics, but they are yet to be understood in terms of their finite sample identification analysis. The technical challenges come from the strong statistical dependency not only between data samples at different time instances but also between elements within each individual sample. In this work, for ARX models with potentially unknown orders, we study how ordinary least squares (OLS) estimator performs in terms of identifying model parameters from data collected from either a single length-T trajectory or N i.i.d. trajectories. Our main results show that as long as the orders of the model are chosen optimistically, i.e., we are learning an over-parameterized model compared to the ground truth ARX, the OLS will converge with the optimal rate O(1/√T) (or O(1/√N)) to the true (low-order) ARX parameters. This occurs without the aid of any regularization, thus is referred to as self-regularization. Our results imply that the oracle knowledge of the true orders and usage of regularizers are not necessary in learning ARX models — over-parameterization is all you need