Sample Complexity of Data-Driven Stochastic LQR with Multiplicative Uncertainty

This paper studies the sample complexity of the stochastic Linear Quadratic Regulator when applied to systems with multiplicative noise. We assume that the covariance of the noise is unknown and estimate it using the sample covariance, which results in a larger closed-loop cost. The main contribution of this paper is then to bound the suboptimality of the methodology — i.e., the difference between the closed-loop cost and the true optimum — and prove that it decreases with 1/N, where N denotes the amount of samples of the noise distribution. Under certain assumptions, the methodology is also applicable when only state trajectories are available. It also generalizes to the case where the mean is unknown and to the distributionally robust case studied in a previous work of the authors. The analysis is mostly based on results from matrix function perturbation analysis.