Questions about choosing metrics

Hi, for a specific machine learning problem, is it possible to me to choose a function, like f(a) to evaluate a, while my loss function is actually generated from g(f(a))? If yes, are there any examples? If not, why I cannot use it? Thanks a lot.

I’m not sure if I have ever seen a use case of an evaluation function f(a) and a loss g(f(a)), but I don’t see why it couldn’t work assuming both methods are differentiable.
Note that you are often using a metric function f(a) and a loss g(a) such as the accuracy as the metric function and e.g. cross-entropy loss as the loss function.