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I am working on a regression problem where I want to model the loss function in a way that it "punishes" to big errors much more than small errors (so I am in the realm of exponential functions) but also in a way that is punishes a negative error much more than a positive error.
So for example:
Prediction off by +4.0: is a problem, but still ok
Prediction off by +0.5: not a big deal
Prediction off by -0.5: is a problem, but still ok
Prediction off by -4.0: is a major problem
My problem is that I cant find a good function to describe this. x squared and so do not have the higher values for negative inputs that I am looking for.
My best workaround for now is to just move the whole function to the right (x-2)^2, but there must be something better?