CoVO-MPC: Theoretical Analysis of Sampling-based MPC and Optimal Covariance Design (2401.07369v1)

Abstract: 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/}.

• The paper presents the first theoretical convergence analysis of MPPI, demonstrating linear convergence in quadratic scenarios.
• It introduces CoVO-MPC, an algorithm that computes optimal covariance matrices to enhance performance across both quadratic and nonlinear cost functions.
• Experimental results on robotic tasks validate significant performance gains, despite higher computational demands mitigated by offline approximations.

Introduction

Sampling-based Model Predictive Control (MPC) has established its utility in handling complex dynamical systems with nonconvex cost functions. One variant, Model Predictive Path Integral Control (MPPI), though empirically successful, exhibits a discernible lack of theoretical grounding particularly in regards to convergence properties and optimal hyperparameter choices. Addressing this gap, this paper presents a novel contribution: the first convergence analysis of MPPI and the introduction of CoVariance-Optimal MPC (CoVO-MPC) – an algorithm that intelligently adjusts the sampling covariance matrix to accelerate convergence rates.

Theoretical Groundwork

The theoretical framework explores the behavior of MPPI across quadratic and non-quadratic cost functions. The authors establish that in a quadratic optimization environment, reflecting time-varying Linear Quadratic Regulator (LQR) systems, MPPI exhibits linear convergence toward the optimal control sequence. Significantly, the convergence rate is a function of the sampling covariance matrix and system parameters.

This finding forms the basis for optimizing the said covariance matrix. Consequently, the proposed CoVO-MPC leverages this knowledge to calculate an optimal covariance matrix that takes into account the dynamics and cost functions of the system, which can be computed either in real time or through offline approximations.

Algorithmic Contributions

CoVO-MPC represents a pragmatic evolution of sampling-based MPC strategies, explicitly serving systems governed by both quadratic costs, as well as strongly convex and other nonlinear costs with non-quadratic features. Emphasized within the work is the construction of an optimal covariance matrix, resulting from meticulous analysis and scalable to general nonlinear environments.

The experimental validation is compelling, with CoVO-MPC outstripping MPPI by substantive margins across an array of robotic tasks. Notably, even when applying offline approximations of the optimal covariance, gains remain significant, underscoring the robustness of the proposed method.

Computational Considerations

While the enhancements in CoVO-MPC's performance are evident, they present an increased computational overhead. Obtaining the Hessian matrix and the optimal covariance matrix requires more processing power, a trade-off that is both acknowledged and quantified in the paper. Nevertheless, CoVO-MPC's offline approximation variant mitigates this to some extent without relinquishing too much performance, hence offering a form of computational compromise.

Conclusion

Overall, this work presents a seminal analysis of MPPI's convergence and introduces a method that capitalizes on theoretical insights to optimize performance. The results point toward significant improvements in both the speed of convergence and the control quality, an advance that paves the way for broader applications and further investigations into MPC algorithms, especially within the shifting landscapes of model-based reinforcement learning.