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I'm not sure if this is what you're looking for, but people have studied consistency of surrogate risk minimization. There, we define a surrogate loss function $L$ and a link $\psi$. We first minimize surrogate loss on our dataset, yielding some surrogate hypothesis $h$. Then we define $f(x) = \psi(h(x))$. This procedure is roughly consistent if, as data $\...

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