  Sampling-based Model Predictive Control (MPC) has been a practical and
effective approach in many domains, notably model-based reinforcement learning,
thanks to its flexibility and parallelizability. Despite its appealing
empirical performance, the theoretical understanding, particularly in terms of
convergence analysis and hyperparameter tuning, remains absent. In this paper,
we characterize the convergence property of a widely used sampling-based MPC
method, Model Predictive Path Integral Control (MPPI). We show that MPPI enjoys
at least linear convergence rates when the optimization is quadratic, which
covers time-varying LQR systems. We then extend to more general nonlinear
systems. Our theoretical analysis directly leads to a novel sampling-based MPC
algorithm, CoVariance-Optimal MPC (CoVo-MPC) that optimally schedules the
sampling covariance to optimize the convergence rate. Empirically, CoVo-MPC
significantly outperforms standard MPPI by 43-54% in both simulations and
real-world quadrotor agile control tasks. Videos and Appendices are available
at \url{https://lecar-lab.github.io/CoVO-MPC/}.
