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Suppose a real-valued function class $\mathcal{F}$ with pseudo dimension less than $d$, I am wondering what is the pseudo dimension of the following function class \begin{equation} \mathcal{F}_2 = \{\min(f_1, f_2), \text{with } f_1,f_2\in \mathcal{F}\} \end{equation} where for any $x$, $\min(f_1, f_2)(x) = \min(f_1(x), f_2(x))$. My intuition tells me that the pseudo dimension of $\mathcal{F}_2$ should be bounded by $2d$, but I just cannot prove it.

Any help is appreciated!

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  • $\begingroup$ Can you elaborate your intuition? The more, the better. If you have tried proving, can you tell where lies the difficulty? $\endgroup$
    – John L.
    May 9, 2022 at 4:55

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