Estimation of multivariate regression

Let $(X,Y)$ be a random vector whose first component takes on values in a measurable space $(\mathfrak{X},\mathfrak{A},\mu)$ with measure $\mu$ and $Y$ be a real-valued random variable. Let
$$ f(x)=E\{Y\mid X=x\} $$
be the regression function of $Y$ on $X$. We consider the problem of estimating $f(x)$ by observations of $n$ independent copies of $(X,Y)$ given $f\inF$, where $F$ is an a priori known set with specified metric characteristics such as $\varepsilon$-entropy or Kolmogorov widths.