Multi-Level Multi-Fidelity Methods for Path Integral and Safe Control
Zhuoyuan Wang, Takashi Tanaka, Yongxin Chen, Yorie Nakahira
TL;DR
The paper tackles risk quantification and safe stochastic control for systems with multiple fidelity models, where naive sampling is prohibitively costly. It proposes a unified multi-level multi-fidelity Monte Carlo (MLMF) estimator that combines time-domain layering (MLMC) and heterogeneous state representations (MFMC), yielding unbiased and consistent estimates under mild conditions. Theoretical results characterize optimal coefficients $a_l^*$ and optimal sample allocations, yielding variance reduction and $O(\epsilon^{-2})$ sample complexity. Empirical results on safety probability estimation and path-integral control demonstrate improved computation-accuracy trade-offs and near-ground-truth performance at substantially lower computational cost.
Abstract
Sampling-based approaches are widely used in systems without analytic models to estimate risk or find optimal control. However, gathering sufficient data in such scenarios can be prohibitively costly. On the other hand, in many situations, low-fidelity models or simulators are available from which samples can be obtained at low cost. In this paper, we propose an efficient approach for risk quantification and path integral control that leverages such data from multiple models with heterogeneous sampling costs. A key technical novelty of our approach is the integration of Multi-level Monte Carlo (MLMC) and Multi-fidelity Monte Carlo (MFMC) that enable data from different time and state representations (system models) to be jointly used to reduce variance and improve sampling efficiency. We also provide theoretical analysis of the proposed method and show that our estimator is unbiased and consistent under mild conditions. Finally, we demonstrate via numerical simulation that the proposed method has improved computation (sampling costs) vs. accuracy trade-offs for risk quantification and path integral control.