I'm aware of techniques to find the minimum of a function $f(x)$ by taking iterative samples, such as Newton's Method.
Suppose, however, that what I can actually observe is $g(x) = f(x) + E$, where $E$ is a random variable of some unknown distribution. Assuming I can measure samples of $g(x)$ at arbitrary $x$, is there a known technique for estimating $x_m$ such that $f(x_m)$ is the minimum?
If this is not possible in the general case, is it possible if I know the distribution of $E$?
One thought I've considered is just using a standard minimization technique, taking multiple samples at each $x$ and computing the mean of $g_i(x)$ to estimate $f(x)$, but I'm wondering if there is a more efficient way to tackle this.