example non-overlap hypothesis set H to make union bound equals vc bound

[wordchao]

consider the union bound, when H consists of only h1 and h2,
Pr[|Ein(g)-Eout(g)|>e]
<=Pr[|Ein(h1)-Eout(h1)|>e or |Ein(h2)-Eout(h2)|>e] （eq.1）
<=Pr[|Ein(h1)-Eout(h1)|>e] + Pr[|Ein(h2)-Eout(h2)|>e] (eq.2）
=2exp(-2Ne^2) (eq.3)

i wonder which kind of h1 and h2 can more "accurately" satisfy the union bound, maybe this means union bound equals vc bound.
in other words, can i find h1 and h2 to let (eq.1)=(eq.2)?

Pr[|Ein(h1)-Eout(h1)|>e or |Ein(h2)-Eout(h2)|>e]
=Pr[|Ein(h1)-Eout(h1)|>e and |Ein(h2)-Eout(h2)|>e] (eq.a)
+Pr[|Ein(h1)-Eout(h1)|>e and |Ein(h2)-Eout(h2)|<=e] (eq.b)
+Pr[|Ein(h1)-Eout(h1)|<=e and |Ein(h2)-Eout(h2)|>e] (eq.c)

if (eq.2)=(eq.1), then,
sup(eq.a)=0
sup(eq.b)=sup(eq.c)=1/2(eq.3)=exp(-2Ne^2)

but i cannot find the example h1&h2, which means h1 and h2 has no overlap when considering the union bound.
can u help? thanks.

[htlin]

One possibility is when which makes always and hence the event of the difference would be of zero probability and the union bound is trivially true. But I guess that's not what you are looking for.

[wordchao]

Hi, htlin. I'm sorry for check the reply late.

Thanks for your proposition and it is a good institution.
I want to find a more general way to build examples but I have not done it.
I tried a small lab,
when the loss function of h1 and h2 are same/or complementary, A=B;
when increase the difference btw loss functions of h1 and h2, the overlap btw A&B increase also.


keep in touch if u would like, i am new in Mearchine learning.
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