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{equation} \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} \end{equation}

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{equation} \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} \end{equation}

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



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