From non-linear least squares to weighted linear least squares

Question

Given an overdetermined linear system $A \in \mathbb{R}^{m \times n}$, $b \in \mathbb{R}^{m \times 1}$. And a non-linear function $f(x)$. Given a non-linear least squares: $$ e^* = \min_g \left\lVert f(A g) - {b}\right\rVert_2^2,$$ where the function $f(x)$ is applied element-wise. What would be the best weight $W$ depending on $b$ for a weighted linear least squares $$ \hat{e} = \min_g \left\lVert W A g - W b\right\rVert_2^2, $$ such that $\min{|e^* - \hat{e}|}$.