How to use derivatives of a function to better estimate its variance over the domain?
I have a scalar smooth function $f(x)$ and a multivariate random variable $x$ with known distribution (e.g. multivariate standard normal with large diagonal sigma). I know that the function f is very peaky in a sense that for majority of x'es its value rarely deviate from the mean, but on a small subset of x'es it attains extreme values. I need a finite sample estimate of the variance of this function over the domain of X. I can sample x, compute f(x) and use a standard formula, but it seems to be not very efficient since the domain of X is large and high dimensional (hundreds of dimensions) and I know that it is mostly constant.
Intuitively, if I encounter a region with low curvature and "close to the mean" value of f(x), I can "mark" it as "not interesting" and explore the rest of X.
Is there a formal way of doing this? Assume for simplicity that f is antisimmetric so f(x) = -f(-x) so we only need to look for maximas of x. One could probably do something like a variational approximation q such that x' ~ q(x'|x) maximizes expectation of f(x') while being not too far from the standard normal, but this way we have a significant chance of greatly underestimating the variance.