I'm looking for an algorithm to find the argmax of an objective function of the form $g(u):=\mathbb{E}_X[f(u,X)]$
The analytical formula of this expected value is intractable - it is impossible to calculate explicitly. However, it is possible to know that $g(u)$ is concave and its maximum $u^*$ is in the interval $]0,1[$.
We also know that $f$ is differentiable in order to the first variable (Hence, the problem is equivalent to find the root of $\mathbb{E}_X[\partial_u f(u,X)]$)
One method that I tried was to generate a random sample of $X$: $x_1, ..., x_n$ and calculate the sequence defined by $h_i = h_{i-1} + \partial_u f(i/n, x_i)$. Hence if $h_k$ is the maximum of this sequence then $k/n$ will be a numerical approximation for the argmax.
However, this method has a bad precision / speed ratio, and I'm not even sure that it is correct. Could you please help me and suggest better algorithms?
Thank you in advance.