Sampling Complexity of Path Integral Methods for Trajectory Optimization

The use of random sampling in decision-making and control has become popular with the ease of access to graphic processing units that can generate and calculate multiple random trajectories for real-time robotic applications. In contrast to sequential optimization, the sampling-based method can take advantage of parallel computing to maintain constant control loop frequencies. Inspired by its wide applicability in robotic applications, we calculate a sampling complexity result applicable to general nonlinear systems considered in the path integral method, which is a sampling-based method. The result determines the required number of samples to satisfy the given error bounds of the estimated control signal from the optimal value with the predefined risk probability. The sampling complexity result shows that the variance of the estimated control value is upper-bounded in terms of the expectation of the cost. Then we apply the result to a linear time-varying dynamical system with quadratic cost and an indicator function cost to avoid constraint sets.