# Why is MSE used over other quadratic loss functions?

So I was wondering, why I have only encountered square loss function also known as MSE. The only nice property of MSE I am so far aware of is its convex nature. But then all equations of the form $$x^{2n}$$ where $$n$$ is an integer belongs to the same family.

My question is what makes MSE the most suitable candidate among this entire family of curves? Why do other curves in the family, even though having steeper slopes, $$(x >1)$$, which might result in better optimisation, not used?

Here is a picture to what I mean where red is $$x^4$$ and green is $$x^2$$:

As you mentioned MSE (aka $$l_2$$-loss) is convex which is a great property in optimization in which one can find a single global optimum. MSE is used in linear and non-linear least squares problems which form the basis of many widely used statistical methods. I would imagine the math and implementation would be more difficult if one would use a higher-order loss (e.g. $$x^3$$) and that would also prove to be futile because MSE already possesses great statistical and optimization properties on its own.
Other interesting losses you might want to read about are $$l_0$$ and $$l_{inf}$$ loss, all of which have their own trade-offs in optimization-sense.