SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 2, Page 309-346, June 2024.
Abstract. Importance sampling is a popular variance reduction method for Monte Carlo estimation, where an evident question is how to design good proposal distributions. While in most cases optimal (zero-variance) estimators are theoretically possible, in practice only suboptimal proposal distributions are available and it can often be observed numerically that those can reduce statistical performance significantly, leading to large relative errors and therefore counteracting the original intention. Previous analysis on importance sampling has often focused on asymptotic arguments that work well in a large deviations regime. In this article, we provide lower and upper bounds on the relative error in a nonasymptotic setting. They depend on the deviation of the actual proposal from optimality, and we thus identify potential robustness issues that importance sampling may have, especially in high dimensions. We particularly focus on path sampling problems for diffusion processes with nonvanishing noise, for which generating good proposals comes with additional technical challenges. We provide numerous numerical examples that support our findings and demonstrate the applicability of the derived bounds.

中文翻译：

---

次优重要性采样的非渐近界

SIAM/ASA 不确定性量化杂志，第 12 卷，第 2 期，第 309-346 页，2024 年 6 月
。摘要。重要性采样是蒙特卡洛估计中流行的方差减少方法，其中一个明显的问题是如何设计良好的提案分布。虽然在大多数情况下，最优（零方差）估计量在理论上是可能的，但在实践中，只能使用次优提议分布，并且通常可以从数值上观察到，这些分布会显着降低统计性能，导致较大的相对误差，从而抵消初衷。先前对重要性采样的分析通常集中于在大偏差情况下效果良好的渐近论证。在本文中，我们提供非渐近设置中相对误差的下限和上限。它们取决于实际建议与最优性的偏差，因此我们确定了重要性采样可能具有的潜在鲁棒性问题，尤其是在高维度中。我们特别关注具有非消失噪声的扩散过程的路径采样问题，为此生成好的建议会带来额外的技术挑战。我们提供了大量的数值例子来支持我们的发现并证明了导出界限的适用性。