# Dual of sum of two sigmoid functions

Suppose we have 2 samples with features $$\mathbf{x}_1,\mathbf{x}_2 \in \mathbb{R}^d$$ and there is an uncertain parameter $$\mathbf{\theta} \in \mathbb{R}^d$$. Also, let's define $$\mu(y)$$ as a sigmoid/logistic function, i.e., $$\mu(y)=1/(1+\exp(-y))$$. I am wondering how it is possible to write the dual of the following problem. $$$$\begin{split} \min_{\theta} \: &\left\{ a_1 \mu(\mathbf{x}_1^T \theta)+a_2\mu(\mathbf{x}_2^T \theta) \right\} \\ & \|\theta-\hat{\theta}\|_{H} \le \gamma \end{split}$$$$

It is assumed that $$a_1,a_2, \gamma>0$$ and $$H$$ is positive definite. I know how to solve it when there is just one sample, but I am confused when there are two samples, and I am wondering how to extend it to the case when there are more samples, $$n>2$$. Is there any condition to write its dual? Since vectors $$\mathbf{x}_i$$ are feature vectors, it is possible to transform them into any interval.

Also, it is possible to find the dual of the following problem? $$$$\begin{split} \min_{\theta} \: &\left\{ a_1 \mu(\mathbf{x}_1^T \theta)-a_2\mu(\mathbf{x}_2^T \theta) \right\} \\ & \|\theta-\hat{\theta}\|_{H} \le \gamma \end{split}$$$$

Finally, is it possible to extend it to other generalized linear models? Thanks