Given a tall matrix $F \in \mathbb{C}^{m \times n}$, where $m > n$, and a non-symmetric matrix $A$ of size $n \times n$, consider the spectral norm
$$ \begin{aligned} \| A - F^* \operatorname{diag}(b) F \|_2 &= \sigma_{\max} \left( A - F^* \operatorname{diag}(b) F \right) \\ &= \sqrt{\lambda_{\max} \left( (A-F^*\operatorname{diag}(b)F)^* (A-F^*\operatorname{diag}(b)F ) \right),} \end{aligned} $$
How do I compute analytically $\nabla_b \|A-F^*\operatorname{diag}(b)F\|_2$, where $b \in \mathbb{C}^{m \times 1}$ is some vector and {$*$} is a sign for conjugate transpose?
I need gradient because I want to find $b$ by minimizing $\|A-F^*\operatorname{diag}(b)F\|_2$ as I would like to find the optimum by using gradient descent. Is it possible?