Beyond exponential tilting: sharp asymptotics and improved importance sampling for rare events

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 ${\epsilon }^{-1\mathrm{/}4}$ in the small-noise limit $\epsilon \downarrow 0$. 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 $\epsilon$. 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.