Function growing faster for negative inputs than for positives
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.