I need to find/approximate the argument that minimizes a stochastic convex function $F(\theta, Z)$: $$ {\arg\min_{\theta}} E_{Z}[ F(\theta, Z) ]$$ Where $Z$ is some random variable (we could assume gaussian) and $\theta$ is a parameter that I can control. The goal is to optimize $\theta$ while sampling $F(\theta, Z)$ as few times as possible, given some termination condition (could be as simple as $F(\theta, z) < a$ for some value of $a$ that I will choose). For my specific problem $\theta \in \mathbf{R^2}$ and I have a "good" initial guess. I also have a coarse model of $F(\theta, Z)$ but would prefer a generic method. One possible assumption would be that $F(\theta, Z)$ could be written on the form $G(\theta) + H(Z)$, where of course $G$ and $H$ are convex.
So far I have only tested random search, with success but it takes many iterations. Computational/implementational complexity is also of essence.
Any suggestions of algorithms/methods or atleast some ideas of what search terms I could use to find similar problems?