Loss Function for $l_0$ norm

Question

Suppose we have $n$ samples of data with feature $Y$. Derive solution for the optimal constant $a^\star_0$

\begin{align*} a^* \in argmin_{a \in \mathcal{R}} \frac{1}{n}\sum_{i=1}^n |Y(i)-a|^0 \end{align*}

I understand this is not differentiable. I am not sure how to go about this. Is this the $l_0$ norm? How can I find the objective function?

Answer 1

The penalization occur whenever $Y(i)$ is not equal to $a$, and the penalization is the same regardless of the magnitude.

Hence the optimal strategy is to set $a$ to be one of the mode and that would minimize the $l_0$ norm.