# Gradient of $-y^T \log \left(f \left(W x ; \gamma, \beta \right) \right)$ w.r.t. $\left\{W, \gamma, \beta\right\}$?

How to compute the gradient of \begin{align} L\left(W, \gamma, \beta\right) := -y^T \log \left(f \left(W x ; \gamma, \beta \right) \right) \end{align} with respect to $$\left\{W, \gamma, \beta\right\}$$, where $$x \in \mathbb{R}^n$$, $$W \in \mathbb{R}^{m \times n}$$, and $$y \in \mathbb{R}^m$$, but $$y_i \in \{0,1\}$$, $$f(z; \gamma, \beta)$$ is parameterized by $$\gamma$$ and $$\beta$$?

The definition of
\eqalign{ f(z; \gamma, \beta) &= \gamma \ \left( z- \mu(z) \right) \ \left( \sigma(z) + \epsilon \right)^{-1/2} + \beta \cr \mu(z) &= \alpha \ 1^Tz \cr \sigma(z) &= \alpha \sum_{k=1}^m \left( z[k] - \mu(z) \right)^2 \equiv \alpha 1^T \left[ \left( z- \mu(z) \right) \odot \left(z - \mu(z) \right) \right]\cr } where $$1^T$$ is a row vector with all ones, $$\odot$$ is an element-wise multiplication, and $$\alpha$$ and $$\epsilon$$ are known scalars.