Overconservativeness of Variance-Based Efficiency Criteria and Probabilistic Efficiency in Rare-Event Simulation

Abstract

In rare-event simulation, an importance sampling (IS) estimator is regarded as efficient if its relative error, namely, the ratio between its standard deviation and mean, is sufficiently controlled. It is widely known that when a rare-event set contains multiple “important regions” encoded by the so-called dominating points, the IS needs to account for all of them via mixing to achieve efficiency. We argue that in typical experiments, missing less significant dominating points may not necessarily cause inefficiency, and the traditional analysis recipe could suffer from intrinsic looseness by using relative error or, in turn, estimation variance as an efficiency criterion. We propose a new efficiency notion, which we call probabilistic efficiency, to tighten this gap. In particular, we show that under the standard Gartner-Ellis large deviations regime, an IS that uses only the most significant dominating points is sufficient to attain this efficiency notion. Our finding is especially relevant in high-dimensional settings where the computational effort to locate all dominating points is enormous. This paper was accepted by Baris Ata, stochastic models and simulation. Funding: This work was supported by the National Science Foundation Division of Information and Intelligent Systems [Grants IIS-1849280 and IIS-1849304], Division of Civil, Mechanical and Manufacturing Innovation [Grant CAREER CMMI-1834710], and Division of Computer and Network Systems [Grant CNS-2047454]. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.4973 .