I'm trying to extend the Rademacher complexity and have the following question:
For $ (v_1,..,v_m) = V \in {\mathbb{R}}^{m} $ , I will be glad to find an upper abound for the Euclidean norm:
$$ ||V|| = \sqrt{\sum_{i=1}^{m}} v_{i}^{2} $$
That somehow involves the average of its coordinates:
$$ \frac{\sum_{i=1}^{m}v_i}{m}$$
More specifically my problem is to bound this expression:
$$ \frac{1}{\sqrt{m}} \mathop{\mathbb{E}}_{ {S}^{'} \sim {\mathcal{D}}^{m}} \Big[ \mathop{sup}_{ w \in {\mathbb{R}}^{n_1} } \Big( ||\hat{f_w}(z)|| \Big) \Big] \leq \beta $$
Such that $\beta\;$ is a function of $\;\; \frac{\sum_{i=1}^{m}\hat{f_w}(z_i)}{m}$
And the notations is as follows: $$ \hat{f_w}(z_i) = \mathop{\mathbb{E}}_{ \gamma \sim q_{w^{*}}(\gamma)} \big[ f((w,\gamma),z_i) \big] $$ $$q_{w^{*}} (\gamma) = \frac{1}{\sqrt{2\pi}} e ^{- \frac{(\gamma - w^*)^2}{2}}$$ $$S = {\lbrace z_i \rbrace}_{i=1}^{m} $$ Thanks a lot for the help.