statistical risk minimization problem

Suppose $x,y$ are related by some (known) statistical model $p(x,y)$, and we want to estimate $p(x,y)$ using a model $\tilde{y}=f(x)$, where $f$ is a member of our hypothesis class $\mathcal{F}$.

Let $\ell (y,\hat{y})$ be our loss function. Then the statistical risk minimization problem is to minimize the expected loss over distribution $p(x,y)$:

The optimal estimator $f^{\ast }$ is the function $f\in \mathcal{F}$ with minimal expected cost over all possible functions $f$.