Beyond exponential tilting: sharp asymptotics and improved importance sampling for rare events
Abstract
A classical approach for importance sampling (IS) of a rare event is to tilt the measure using its large deviation minimizer, thereby shifting the proposal mean towards the event set. While this tilt is log-efficient, we show that the coefficient of variation of the resulting estimator still grows like ε−1/4 in the small-noise limit ε↓0. Motivated by sharp large deviation asymptotics and the local geometry of the event set at the minimizer, we propose improvements using a control variate or an anisotropic, non-Gaussian proposal. We prove, up to truncations that hold with high probability, and confirm empirically in high-dimensional examples, that the former bounds the coefficient of variation and the latter makes it vanish, so that a bounded number of samples suffices. Our analysis addresses both the coefficient of variation of the different proposals and their necessary and sufficient sample size based on the Kullback--Leibler divergence of the IS target from the proposal; both yield the same rates in ε. We demonstrate the resulting samplers, including a variant that blends the classical and the new proposal across noise levels, for a stochastic Korteweg--de Vries equation with thousands of random inputs.
Related
- Stretched exponential asymptotics for bases of triangular bootstrap percolation
- Multiple Importance Sampling for Guided Speculative Inference
- Post-stratified importance sampling
- Repulsive normalizing flow mixtures for adaptive importance sampling: reliability analysis of complex systems
- Amortized ratio-estimation importance sampling and localized simulation-based calibration for intractable likelihoods
BibTeX
@misc{li2026beyond,
title = {{Beyond exponential tilting: sharp asymptotics and improved importance sampling for rare events}},
author = {Mingxin Li and Hans Reimann and Timo Schorlepp and Georg Stadler},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27434},
eprint = {2610.27434},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27434},
}
RIS
TY - UNPB
TI - Beyond exponential tilting: sharp asymptotics and improved importance sampling for rare events
AU - Mingxin Li
AU - Hans Reimann
AU - Timo Schorlepp
AU - Georg Stadler
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27434
UR - https://arxiv.org/abs/2610.27434
AB - A classical approach for importance sampling (IS) of a rare event is to tilt the measure using its large deviation minimizer, thereby shifting the proposal mean towards the event set. While this tilt is log-efficient, we show that the coefficient of variation of the resulting estimator still grows like ε^−1/4 in the small-noise limit ε↓0. Motivated by sharp large deviation asymptotics and the local geometry of the event set at the minimizer, we propose improvements using a control variate or an anisotropic, non-Gaussian proposal. We prove, up to truncations that hold with high probability, and confirm empirically in high-dimensional examples, that the former bounds the coefficient of variation and the latter makes it vanish, so that a bounded number of samples suffices. Our analysis addresses both the coefficient of variation of the different proposals and their necessary and sufficient sample size based on the Kullback--Leibler divergence of the IS target from the proposal; both yield the same rates in ε. We demonstrate the resulting samplers, including a variant that blends the classical and the new proposal across noise levels, for a stochastic Korteweg--de Vries equation with thousands of random inputs.
ER -