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Optimality and Suboptimality of MPPI Control in Stochastic and Deterministic Settings

Hannes Homburger, Florian Messerer, Moritz Diehl, Johannes Reuter

TL;DR

The paper analyzes Model Predictive Path Integral (MPPI) control from a stochastic optimal control perspective, showing that standard MPPI solves three related OCPs—CLS-OCP, OLS-OCP, and DET-OCP—under an input-noise interpretation. It introduces a beta-scaling parameter that tunes sampling uncertainty and proves that, in smooth unconstrained problems, the suboptimality of the MPPI control scales as $O(\beta^2)$ while the suboptimality of the value function scales as $O(\beta^4)$. The authors derive a Laplace-method-based convergence result for the deterministic case, showing that MPPI trajectories converge to the true DET-OCP solution as $\beta \to 0$. They also discuss computational aspects, including Monte Carlo sampling, importance sampling corrections, and GPU-friendly implementations, and illustrate the theory with numerical experiments. These results provide principled guidance for tuning MPPI hyperparameters in robotics and reinforcement learning applications.

Abstract

Model predictive path integral (MPPI) control has recently received a lot of attention, especially in the robotics and reinforcement learning communities. This letter aims to make the MPPI control framework more accessible to the optimal control community. We present three classes of optimal control problems and their solutions by MPPI. Further, we investigate the suboptimality of MPPI to general deterministic nonlinear discrete-time systems. Here, suboptimality is defined as the deviation between the control provided by MPPI and the optimal solution to the deterministic optimal control problem. Our findings are that in a smooth and unconstrained setting, the growth of suboptimality in the control input trajectory is second-order with the scaling of uncertainty. The results indicate that the suboptimality of the MPPI solution can be modulated by appropriately tuning the hyperparameters. We illustrate our findings using numerical examples.

Optimality and Suboptimality of MPPI Control in Stochastic and Deterministic Settings

TL;DR

The paper analyzes Model Predictive Path Integral (MPPI) control from a stochastic optimal control perspective, showing that standard MPPI solves three related OCPs—CLS-OCP, OLS-OCP, and DET-OCP—under an input-noise interpretation. It introduces a beta-scaling parameter that tunes sampling uncertainty and proves that, in smooth unconstrained problems, the suboptimality of the MPPI control scales as while the suboptimality of the value function scales as . The authors derive a Laplace-method-based convergence result for the deterministic case, showing that MPPI trajectories converge to the true DET-OCP solution as . They also discuss computational aspects, including Monte Carlo sampling, importance sampling corrections, and GPU-friendly implementations, and illustrate the theory with numerical experiments. These results provide principled guidance for tuning MPPI hyperparameters in robotics and reinforcement learning applications.

Abstract

Model predictive path integral (MPPI) control has recently received a lot of attention, especially in the robotics and reinforcement learning communities. This letter aims to make the MPPI control framework more accessible to the optimal control community. We present three classes of optimal control problems and their solutions by MPPI. Further, we investigate the suboptimality of MPPI to general deterministic nonlinear discrete-time systems. Here, suboptimality is defined as the deviation between the control provided by MPPI and the optimal solution to the deterministic optimal control problem. Our findings are that in a smooth and unconstrained setting, the growth of suboptimality in the control input trajectory is second-order with the scaling of uncertainty. The results indicate that the suboptimality of the MPPI solution can be modulated by appropriately tuning the hyperparameters. We illustrate our findings using numerical examples.

Paper Structure

This paper contains 16 sections, 4 theorems, 37 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction and Overview
  2. Contribution and Outline
  3. Notation and Preliminaries
  4. Stochastic Optimal Control Problems
  5. Closed-Loop Stochastic Optimal Control
  6. Open-Loop Stochastic Optimal Control
  7. Deterministic Optimal Control
  8. MPPI Solution Methods
  9. MPPI Control for the CLS-OCP
  10. MPPI Control for the OLS-OCP
  11. MPPI Control for the D-OCP
  12. MPPI Control Algorithms
  13. Investigation of Suboptimality
  14. Alternatives and Computational Effort
  15. Numerical Experiments
  16. ...and 1 more sections

Key Result

Proposition 1

We assume the dynamics eq_affin_system, $w$ i.i.d., the stage cost eq_stage_cost, and Assumption assumption holds. Then an equivalent OCP exists, which is specified by the dynamics and the corresponding overall cost eq_overall_cost.

Figures (3)

  • Figure 1: PDF of optimal distribution $\mathbb{Q}^\star_\beta(W)$ for different $\beta$. The two additional interior plots show zoomed sections of the same PDF.
  • Figure 2: Suboptimality of the MPPI solution to the DET-OCP of the controls (gold) and the value function (blue).
  • Figure 3: Numerical solutions to the three different problems DET-OCP, OLS-OCP, and CLS-OCP (black), all with overall cost function \ref{['eq_ex2']} based on input-affine dynamics $f_\mathrm{af}$ (top) and nonlinear dynamics $f_\mathrm{nl}$ (bottom) alongside the iterates of deterministic MPPI.

Theorems & Definitions (7)

  • Proposition 1
  • proof
  • Lemma 1: Erdélyi’s formulation of Laplace's classical method, adapted from Nemes.2013
  • Theorem 1
  • proof
  • Corollary 1.1
  • proof