I'm trying to compute the gradient of a complex loss function but I'm stuck on calculating the derivative of part of the expression,namely:
with respect to $w_i$
$$\sum_{n=1}^N(y^n -w_0 - \sum_{j=1}^p A_{nj}w_j)^2$$
A - design matrix; w - vector; y - vector; $w_0$ - first element of w
Edit Where it said k it should have been n - now fixed
For $i\in\{1,\ldots,p\}$:
$$\begin{align*}&\phantom{=}\ \, \frac{\partial}{\partial w_i}\sum_{n=1}^N\left[\left(y^n -w_0 - \sum_{j=1}^p A_{nj}w_j\right)^2\right]\\ &=\sum_{n=1}^N\frac{\partial}{\partial w_i}\left[\left(y^n -w_0 - \sum_{j=1}^p A_{nj}w_j\right)^2\right]\\ &=\sum_{n=1}^N\left[2\left(y^n -w_0 - \sum_{j=1}^p A_{nj}w_j\right)\frac{\partial}{\partial w_i}\left(y^n -w_0 - \sum_{j=1}^p A_{nj}w_j\right)\right]\\ &=\sum_{n=1}^N\left[2\left(y^n -w_0 - \sum_{j=1}^p A_{nj}w_j\right)\left( -\sum_{j=1}^p A_{nj}\frac{\partial w_j}{\partial w_i}\right)\right]\\ &=\sum_{n=1}^N\left[2\left(y^n -w_0 - \sum_{j=1}^p A_{nj}w_j\right)\left( - A_{ni}\right)\right]\\ &=-2\sum_{n=1}^N\left[A_{ni}\left(y^n -w_0 - \sum_{j=1}^p A_{nj}w_j\right)\right] \end{align*}$$