Suppose we have a function $f(x)$ that we can only observe through some noise. We can not compute $f(x)$ directly, only $f(x) + \eta$ where $\eta$ is some random noise. (In practice: I compute $f(x)$ using some Monte Carlo method.)
What methods are available for finding roots of $f$, i.e. computing $x$ so that $f(x) = 0$?
I am looking for methods which minimize the number of evaluations needed for $f(x)+\eta$, as this is computationally expensive.
I am particularly interested in methods that generalize to multiple dimensions (i.e. solve $f(x,y) = 0, g(x,y) = 0$).
I'm also interested in methods that can make use of some information about the variance of $\eta$, as an estimate of this may be available when computing $f(x)$ using MCMC.