Can Gradient Descent Provably Learn Linear Dynamic Systems?

We study the learning ability of linear recurrent neural networks with gradient descent. We prove the first theoretical guarantee on linear RNNs with Gradient Descent to learn any stable linear dynamic system. We show that despite the non-convexity of the optimization loss if the width of the RNN is large enough (and the required width in hidden layers does not rely on the length of the input sequence), a linear RNN can provably learn any stable linear dynamic system with the sample and time complexity polynomial in $\frac{1}{1-\rho_C}$ where $\rho_C$ is roughly the spectral radius of the stable system. Our results provide the first theoretical guarantee to learn a linear RNN and demonstrate how can the recurrent structure help to learn a dynamic system.