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Error Analysis of Sampling Algorithms for Approximating Stochastic Optimal Control

Anant A. Joshi, Amirhossein Taghvaei, Prashant G. Mehta

TL;DR

This paper analyzes the one-step error of two sampling-based approaches for stochastic optimal control—Model Predictive Path Integral (MPPI) and an ensemble Kalman filter–inspired Interacting Particle System (IPS)—through a Gibbs variational lens. For a simple single-stage problem with quadratic costs, it derives closed-form expressions for the approximation error as a function of dimension $d$ and sample size $N$, and shows MPPI suffers from an exponential-in-dimension growth in error whereas IPS scales only linearly with $d$. The results are supported by numerical simulations that confirm the theoretical scaling, highlighting a significant dimension-dependent trade-off between MPPI and IPS. The Gibbs framework also clarifies the connections to filtering and duality with posterior inference, providing a unified view of how control laws can be approximated via sampling. Overall, the work offers concrete guidance on method choice and scaling behavior for high-dimensional stochastic control tasks.

Abstract

This paper is concerned with the error analysis of two types of sampling algorithms, namely model predictive path integral (MPPI) and an interacting particle system (\IPS) algorithm, that have been proposed in the literature for numerical approximation of the stochastic optimal control. The analysis is presented through the lens of Gibbs variational principle. For an illustrative example of a single-stage stochastic optimal control problem, analytical expressions for approximation error and scaling laws, with respect to the state dimension and sample size, are derived. The analytical results are illustrated with numerical simulations.

Error Analysis of Sampling Algorithms for Approximating Stochastic Optimal Control

TL;DR

This paper analyzes the one-step error of two sampling-based approaches for stochastic optimal control—Model Predictive Path Integral (MPPI) and an ensemble Kalman filter–inspired Interacting Particle System (IPS)—through a Gibbs variational lens. For a simple single-stage problem with quadratic costs, it derives closed-form expressions for the approximation error as a function of dimension and sample size , and shows MPPI suffers from an exponential-in-dimension growth in error whereas IPS scales only linearly with . The results are supported by numerical simulations that confirm the theoretical scaling, highlighting a significant dimension-dependent trade-off between MPPI and IPS. The Gibbs framework also clarifies the connections to filtering and duality with posterior inference, providing a unified view of how control laws can be approximated via sampling. Overall, the work offers concrete guidance on method choice and scaling behavior for high-dimensional stochastic control tasks.

Abstract

This paper is concerned with the error analysis of two types of sampling algorithms, namely model predictive path integral (MPPI) and an interacting particle system (\IPS) algorithm, that have been proposed in the literature for numerical approximation of the stochastic optimal control. The analysis is presented through the lens of Gibbs variational principle. For an illustrative example of a single-stage stochastic optimal control problem, analytical expressions for approximation error and scaling laws, with respect to the state dimension and sample size, are derived. The analytical results are illustrated with numerical simulations.

Paper Structure

This paper contains 23 sections, 12 theorems, 56 equations, 2 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Aims and original contributions of this paper
  3. Related literature
  4. Outline of the paper
  5. Relation of $\mathsf{Gibbs}(\gamma,c)$ to filtering and control
  6. Relation of $\mathsf{Gibbs}(\gamma,c)$ to optimal control
  7. Relation of $\mathsf{Gibbs}(\gamma,c)$ to filtering, and duality
  8. Sampling algorithms to find optimal control
  9. MPPI
  10. Interacting particle system (IPS)
  11. Scaling of error with dimensions
  12. Numerical simulation
  13. Conclusion and future work
  14. Gibbs variational problem
  15. Details of Sec. \ref{['sec:control']}
  16. ...and 8 more sections

Key Result

Proposition 1

Consider eq:DTOC1-simple with $c(x):= \frac{1}{2} |x|^2$. For $U=u \in \mathcal{U}^{\mathrm{det}}$,

Figures (2)

  • Figure 1: Numerical illustration of Prop. \ref{['prop:onestep']}, with simulation details provided in Sec. \ref{['sec:sim']}. The y-axis shows the mean square error in estimating the optimal control (m.s.e.) using the MPPI and IPS algorithm respectively, and the x-axis is the state dimension $d$. The m.s.e. in MPPI grows exponentially in $d$, and m.s.e. in IPS grows affinely in $d$.
  • Figure 2: Pictorial description of \ref{['eq:gibbs-measure-control']}

Theorems & Definitions (27)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 3
  • Proposition 4
  • ...and 17 more