Function growing faster for negative inputs than for positives I am working on a regression problem where I want to model the loss function in a way that it &quot;punishes&quot; 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?
This should be possible using a piecewise exponential loss, something like this:
$
f(x) = 
\begin{cases}
x^2 &amp; x &lt; 0\\
\lambda x^2 &amp; x \ge 0
\end{cases}
$
with $0 &lt; \lambda &lt; 1$. A $\lambda$ of around 0.02 should roughly give you the scale you want.