[Paper Review] Cross-entropy optimisation of importance sampling parameters for statistical model checking

Cyrille Jégourel, Axel Legay+1 more|arXiv (Cornell University)|Jan 25, 2012

Software Reliability and Analysis Research22 references4 citations

TL;DR

This paper proposes a cross-entropy optimisation method to automatically tune importance sampling parameters in statistical model checking, using a low-dimensional parametrisation to avoid full state-space exploration. The approach achieves over 10⁵ variance reduction in rare-event estimation, significantly improving simulation efficiency for complex systems in reliability and systems biology.

ABSTRACT

Statistical model checking avoids the exponential growth of states associated with probabilistic model checking by estimating properties from multiple executions of a system and by giving results within confidence bounds. Rare properties are often very important but pose a particular challenge for simulation-based approaches, hence a key objective under these circumstances is to reduce the number and length of simulations necessary to produce a given level of confidence. Importance sampling is a well-established technique that achieves this, however to maintain the advantages of statistical model checking it is necessary to find good importance sampling distributions without considering the entire state space. Motivated by the above, we present a simple algorithm that uses the notion of cross-entropy to find the optimal parameters for an importance sampling distribution. In contrast to previous work, our algorithm uses a low dimensional vector of parameters to define this distribution and thus avoids the often intractable explicit representation of a transition matrix. We show that our parametrisation leads to a unique optimum and can produce many orders of magnitude improvement in simulation efficiency. We demonstrate the efficacy of our methodology by applying it to models from reliability engineering and biochemistry.

Motivation & Objective

• To address the challenge of efficiently estimating rare properties in large-scale stochastic systems where standard Monte Carlo simulation is impractical due to high variance.
• To develop a parameterised importance sampling distribution that avoids explicit representation of the full transition matrix, enabling tractable optimisation.
• To provide a scalable, automatic method for variance reduction in statistical model checking using cross-entropy minimisation on a low-dimensional parameter vector.
• To demonstrate the method’s effectiveness on real-world models from reliability engineering and systems biology with high state-space complexity.

Proposed method

• The method uses a parametrised importance sampling distribution defined by a low-dimensional vector of parameters, representing transition rates or probabilities in guarded command models.
• It applies cross-entropy optimisation to iteratively update these parameters by minimising the estimated variance of the simulation results.
• The algorithm uses a smoothing strategy to prevent parameter explosion and ensure convergence, particularly for rare transitions.
• The parametrisation is designed to be unique and convex, guaranteeing a single global optimum for the optimisation process.
• The method relies on sampling execution traces under the current parameterised distribution and updating parameters based on the cross-entropy of the empirical distribution of successful paths.
• The algorithm is applied iteratively until convergence, with performance monitored via estimated probability and sample variance.

Experimental results

Research questions

• RQ1 Can a low-dimensional parametrisation of importance sampling distributions be used to efficiently estimate rare properties in large stochastic models?
• RQ2 Does cross-entropy optimisation of these parameters lead to a unique and globally optimal solution?
• RQ3 To what extent can this method reduce variance in statistical model checking compared to standard Monte Carlo simulation?
• RQ4 How well does the method scale and perform on real-world models with high state-space complexity?

Key findings

• The proposed parametrisation leads to a unique global optimum, ensuring stable and reliable convergence of the optimisation process.
• The method achieved a variance reduction of more than 10⁵ in the reliability model, significantly improving simulation efficiency.
• The estimated probability converged to within 1.7% of the true value (computed by PRISM), validating the accuracy of the approach.
• The algorithm successfully identified and down-weighted less relevant repair transitions, focusing sampling on direct failure paths while still considering intermediate repairs.
• The method demonstrated strong performance on a systems biology model with a massive state space, confirming its applicability to complex, real-world systems.