Discrete-Time Stochastic LQR via Path Integral Control and Its Sample Complexity Analysis

In this paper, we derive the path integral control algorithm to solve a discrete-time stochastic Linear Quadratic Regulator (LQR) problem and carry out its sample complexity analysis. While the stochastic LQR problem can be efficiently solved by the standard backward Riccati recursion, our primary focus in this paper is to establish the foundation for a sample complexity analysis of the path integral method when the analytical expressions of optimal control law and the cost are available. Specifically, we derive a bound on the error between the optimal LQR input and the input computed by the path integral method as a function of the sample size. Our analysis reveals that the sample size required exhibits a logarithmic dependence on the dimension of the control input. Lastly, we formulate a chance-constrained optimization problem whose solution quantifies the worst-case control performance of the path integral approach.