How to find robust counterpart of sum of logit functions?

Suppose function $$\mu_i(y):\mathbb{R} \rightarrow \mathbb{R}$$ is a logit function, $$\mu_i(y)=1/(1+\exp(-y))$$. Also, we assume that $$\mathbf{x}_i\in \mathbb{R}^d$$ and $$\theta \in \mathbb{R}^d$$. I am wondering how it is possible to find the robust counterpart of the following problem? $$$$\begin{split} \min_{\theta \in \mathbb{R}^d} &\left\{\sum_{i=1}^{n} a_i \mu_i(\mathbf{x}^T \theta)\: \:\: s.t. \:\:\:\|\theta-\hat{\theta}\|_H \le \gamma \right\} \ge b \end{split}$$$$

where $$H$$ is a positive definite matrix and $$a_i, \gamma, b$$ are positive numbers. If it is not possible to find an exact robust counterpart, I would be thankful for any idea about how to find a conservative approximation.

I crossposted to Math.SE as well: Dual of sum of two sigmoid functions.

• You're looking for the convex conjugate (Fenchel Dual) of this problem? Your logit function is close to some found in the table here: en.wikipedia.org/wiki/Convex_conjugate Commented Oct 27, 2022 at 17:00