if we have this function:
$$L_1(\alpha^{(1)},...,\alpha^{(k)},\beta_1,...,\beta_k)\doteq \frac{1}{2m}*\sum_{i=1}^{m}(f(\widetilde{x}_i)-\widetilde{y}_i)^2 + \gamma *\sum_{k=1}^{K}(\left \| \alpha^{(k)} \right \|_2^{2} + \beta_k^{2})$$
where
$$f_T(x)=T*\ln(\sum_{k=1}^{K}(\exp(<\alpha^{(k)},x/T>)+\beta_k/T)$$
with $\alpha^{(k)}\in\mathbb{R}^{n}$, $\beta_{k}\in\mathbb{R}$, k=1,...,K, $\widetilde{x}_i=\frac{x_i}{T}$, $\widetilde{y}_i=\frac{y_i}{T}$ and $<a,b> \doteq a^Tb$ (which denotes the standard inner product)
how can I write these
$$\alpha^{(k)}(t+1)=\alpha^{(k)}(t)-\eta \nabla_{\alpha^{(k)}(t)}L_1$$
$$\beta_{k}(t+1)=\beta_{k}(t)-\eta\frac{\partial L_1}{\partial \beta_{k}(t))}$$
expliciting the gradients and the derivatives?
In $L_1$ function is assumed T=1.
I don't know how calculate the gradient because $\alpha^{(K)}$ is a vector and changes his value depending on k. Similar problem with $\beta_k$.
Thanks a lot if you can help me!