I came across a note where it does something like the following:
\begin{align} l &= \min \{H, \max_{j\in [M]}\phi^\top(\theta + \xi^{j}) \} - \delta \\ & \geq [\max_{j\in [M]}\phi^\top(\theta + \xi^{j}) - \gamma ] - [\delta - \gamma], \end{align}
where $H \in \mathbb{N}$, $\phi \in \mathbb{R}^d$, $\delta \in \mathbb{R}$, $ \gamma \in \mathbb{R}$. Also $\xi^j \in \mathbb{R}^d$ which are noise sampled from a normal distribution. As a reasoning for the inequality it just says, "$\min$ is a non-expansive function." I am not sure how the argument is working. It seems wrong to me. Can someone shed light regardless of the above being right or wrong?