  Importance sampling is a popular variance reduction method for Monte Carlo
estimation, where a notorious 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. In this article, we provide
nonasymptotic lower and upper bounds on the relative error in importance
sampling that 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 focus on path sampling problems for
diffusion processes, for which generating good proposals comes with additional
technical challenges, and we provide numerous numerical examples that support
our findings.
