Robust Importance Sampling for Rare Events via Constrained Gaussian Mixtures

We study estimating rare-event probabilities $I = \mathbb{P}(g(\mathbf{X}) > \gamma)$ with $\mathbf{X} \sim \mathcal{N}(\mu, \Sigma)$ and general $g : \mathbb{R}^d \to \mathbb{R}$. We address this problem through importance sampling, and propose a framework that substantially improves efficiency and robustness over baselines such as crude Monte Carlo, adaptive cross-entropy, variational-inference-based methods (including forward- and reverse-KL approaches), as well as Safe-ICE, Subset Simulation, and Sequential Monte Carlo, *drawing* on ideas from both cross-entropy methods for rare-event estimation and cross-entropy methods for optimization. The key contribution has two parts: first, we separate the problem into *coverage*, to overcome the cold-start barrier, and *fitting*, to refine proposals once a meaningful signal is available; second, we constrain the proposal family in a way that provably guarantees finite-variance importance sampling, supported by a theoretical result (since coverage alone is not sufficient --- without safeguards, importance sampling may still suffer from infinite variance). Together, these ingredients yield proposals that are both expressive and stable. Extensive experiments demonstrate significant variance reduction, strong robustness across diverse benchmarks, and favorable cost--efficiency trade-offs, with the proposed approach often outperforming these baselines, particularly in high-dimensional and multimodal settings where competing methods frequently become unstable or fail.