Importance Sampling for Statistical Certification of Viable Initial Sets

Abstract: We study the problem of statistically certifying viable initial sets (VISs) -- sets of initial conditions whose trajectories satisfy a given control specification. While VISs can be obtained from model-based methods, these methods typically rely on simplified models. We propose a simulation-based framework to certify VISs by estimating the probability of specification violations under a high-fidelity or black-box model. Since detecting these violations may be challenging due to their scarcity, we propose a sample-efficient framework that leverages importance sampling to target high-risk regions. We derive an empirical Bernstein inequality for weighted random variables, enabling finite-sample guarantees for importance sampling estimators. We demonstrate the effectiveness of the proposed approach on two systems and show improved convergence of the resulting bounds on an Adaptive Cruise Control benchmark.

• The paper introduces an adaptive importance sampling method that statistically certifies viable initial sets in controlled systems.
• It employs defensive mixtures and Gaussian process regression to precisely capture rare failure events in high-dimensional spaces.
• The approach achieves provable PAC-style guarantees with an O(N⁻¹) convergence rate, enhancing efficiency in safety validation.

Sample-Efficient Certification of Viable Initial Sets via Importance Sampling

Problem Formulation and Motivation

The statistical certification of viable initial sets (VISs) for controlled dynamical systems is of critical importance for ensuring compliance with safety or performance specifications, particularly in the context of high-fidelity or black-box system models where analytical characterizations are infeasible. VISs, the sets of initial conditions guaranteeing satisfaction of control specifications, are traditionally obtained through model-based approaches such as controlled invariance or reachability analysis. However, these approaches are limited by the accuracy and tractability of simplified or surrogate models, potentially introducing conservatism or unsafe approximations when deployed in real-world systems.

This work addresses the gap between theoretical VIS computations and practical validation by proposing a data-driven, simulation-based methodology for certifying candidate VISs. The framework focuses on quantifying the probability of specification violations under the true system dynamics, thereby characterizing the safety and performance of candidate sets more robustly.

Methodological Advances

Importance Sampling for Rare-Event Certification

Monte Carlo-based simulation is a conventional approach for safety validation, but it suffers from intractable sample complexity when estimating probabilities of rare failure events. To alleviate this, the paper introduces an importance sampling (IS) scheme that adaptively biases the sampling distribution towards high-risk regions within the candidate VIS. The IS estimator utilizes defensive mixtures, combining a nominal distribution with a surrogate focused on the failure-prone set. This mixture ensures bounded importance weights and improves coverage of rare failure scenarios.

The failure-prone set itself is characterized using Gaussian process (GP) regression with a boundary-focused acquisition strategy. Convex hulls or unions of polytopes are used to approximate the level set corresponding to failures, supporting tractable surrogate distributions even in high-dimensional state spaces.

PAC-style Statistical Guarantees

Standard confidence bounds for failure probability, e.g., binomial tail inversion, do not readily accommodate the weighted samples of IS estimators. The paper develops a PAC (Probably Approximately Correct) bound based on an empirical Bernstein inequality, generalized to bounded, weighted random variables. The resultant concentration inequality provides explicit finite-sample confidence levels for IS-based estimates, leveraging empirical variance for tighter, adaptive guarantees.

The importance-weighted estimator converges at a rate $O(N^{-1})$, significantly faster than the $O(N^{-1/2})$ rate associated with standard Monte Carlo point estimates, especially as the surrogate distribution concentrates on failures. This sample efficiency is demonstrated empirically and analytically.

Empirical Evaluation

The framework is evaluated on two benchmarks:

• Adaptive Cruise Control (ACC): A low-dimensional system with analytically characterized candidate VIS. The IS estimator rapidly converges to the true failure probability with low variance, outperforming the binomial tail inversion.
• Quadrotor Flight Control: A high-dimensional, PD-controlled 12-state system with signal temporal logic specifications. Failure regions are identified via GP modeling, and IS yields conservative and tight failure probability bounds. The analysis illustrates the robustness of defensive mixtures—preventing over-reliance on potentially inaccurate surrogates—while maintaining substantial gains in convergence and coverage over baseline approaches.

Empirical results confirm that the IS-PAC estimator provides accurate certification and improved sample efficiency in both low- and high-dimensional contexts. Sensitivity analyses on the mixture coefficient $\alpha$ demonstrate the trade-off between efficiency and robustness, highlighting the necessity of conservative parameterization when surrogate models are uncertain.

Theoretical and Practical Implications

The proposed framework decouples the certification of VISs from the limitations of model-based methods by leveraging high-fidelity simulation and adaptive exploration. The importance sampling methodology, coupled with concentration bounds for weighted statistics, provides formal finite-sample guarantees, supporting certification for complex, real-world systems where model-based approaches are insufficient.

Practically, this enables the use of candidate VISs as state constraints or safety envelopes in MPC and runtime monitoring, even when underlying dynamics are proprietary or non-analytical. The approach is scalable, robust to dimensionality, and generalizable across control domains—provided tractability of density functions and set representations.

Theoretically, the generalization of the empirical Bernstein inequality to weighted random variables and the demonstration of improved sample complexity offer a rigorous foundation for statistical certification in safety-critical applications. Future extensions may include adaptive mixture design, more expressive surrogate distributions, and online certification in nonstationary settings.

Conclusion

This paper introduces a principled and sample-efficient framework for statistical certification of viable initial sets, integrating importance sampling, adaptive surrogate modeling, and PAC-style concentration guarantees. The empirical and theoretical analysis demonstrates significant improvements in convergence, robustness, and tractability for failure probability estimation in complex dynamical systems. The methodology advances simulation-based certification and lays groundwork for scalable safety assurance in autonomous and cyber-physical control settings (2604.02939).