I am looking for a bound on the empirical Rademacher complexity of the following class:
$G=\left\{x \rightarrow \frac{h^T f(x)}{\|h\|\_2 \cdot \|f(x)\|\_2} : h\in R^d, f()=(f\_1(),\ldots,f\_d()), f\_j \in F \right\}$, where $F$ is some other function class. 


$$\hat{R}\_N(G) = E\_\sigma \sup\_{h\in R^d, \forall j, f\_j\in F}
\frac{1}{N} \left[ \sum\_{n=1}^N \sigma\_n \frac{h^T}{\|h\|}\cdot \frac{f(x\_n)}{\|f(x\_n)\|}\right] \le \text{?}$$


where $\sigma=(\sigma\_1,\ldots,\sigma\_N)$ are i.i.d. Rademacher variables.


The special case of $F=\{x\rightarrow x-c : c\in R^d\}$ would also be of interest.


Is it possible to get better than assuming that $\min\_n \|f(x\_n)\}\|$ is bounded away from 0 and have $O(\sqrt{d}\hat{R}\_N(F))$?



Rademacher complexity of composition of functions
Assume that $h$ belongs to a set $H$, where each vector $v\in H$ satisfies
$\sum\_{i=1}^d |v\_i|\le1$. Then any expression of the form
$ h\cdot \bar f$, where $\bar f=(f\_1,\ldots,f\_d)\in F^d$ belongs to the *absolute convex hull* of $F$. Thus, under the above assumption on $H$, we have
$$ 
\frac1n\mathbb{E}\_\sigma \sup\_{h\in H,(\bar f=f\_1,\ldots,f\_d)\in F^d}\sum\_{i=1}^n \sigma\_i h\cdot \bar f(x\_i)
\le r\_F,
$$
where $r\_F$ is the Rademacher complexity of $F$. For a proof of this claim, see
Theorem 3.3 here <http://www.esaim-ps.org/articles/ps/pdf/2005/01/ps0420.pdf>


Note that the bound is dimension-free (i.e., independent of $d$) and only depends on the $\ell\_1$ norm of $h$ --- and, of course, the Rademacher complexity of $F$.


