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 $O(\sqrt{d}\hat{R}_N(F))$?

insidethe expectation over $\sigma$. You have it, incorrectly, on the outside. $\endgroup$ – Aryeh Kontorovich Apr 15 '17 at 22:44