Function growing faster for negative inputs than for positives

Question

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?

Answer 1

This should be possible using a piecewise exponential loss, something like this:

$ f(x) = \begin{cases} x^2 & x < 0\\ \lambda x^2 & x \ge 0 \end{cases} $

with $0 < \lambda < 1$. A $\lambda$ of around 0.02 should roughly give you the scale you want.