The function is $f(w)= \displaystyle\frac{1}{n} \displaystyle\sum_{i=1}^n\left ( N \left ( \displaystyle\sum_{j=1}^{7} x _{ij}w_j\right) - G_i\right )^2$ where N is the sigmoid function. w is a vector with entries $w_1,w_2,\ldots,w_7 $
my attempt:
First, we have to find the gradient of the function:
$\nabla f(w) = \displaystyle\frac{2}{n}\displaystyle\sum_{i=1}^n\left ( N \left ( \displaystyle\sum_{j=1}^{7} x _{ij}w_j\right) - G_i\right ) \left( N '\left ( \displaystyle\sum_{j=1}^{n} x _{ij}w_j\right)\right)\displaystyle\sum_{j=1}^7 x_{ij}$ and $N'(x)= N(x)(1-N(x))$
$| \nabla f(w)-\nabla f(w')|\leq \displaystyle\frac{2}{n}\displaystyle\sum_{j=1}^7 x_{ij}\left( \Bigg|\displaystyle\sum_{i=1}^n\left ( N \left ( \displaystyle\sum_{j=1}^{7} x _{ij}w_j\right) - G_i\right ) \left( N '\left ( \displaystyle\sum_{j=1}^{n} x _{ij}w_j\right)\right)-\\ \displaystyle\sum_{i=1}^n\left ( N \left ( \displaystyle\sum_{j=1}^{7} x _{ij}w_j'\right) - G_i\right ) \left( N '\left ( \displaystyle\sum_{j=1}^{n} x _{ij}w_j'\right)\right)\Bigg| \right)$
I am stuck here. Anyone help??