prove that mean absolute error is convex

Hello,

I am struggling a bit to prove that mean absolute error is convex (as asked in lecture 1d slide 7), Can someone please show how?

Thanks!

Hey there,

You need two things:

  1. The sum of convex functions is a convex function.
  2. The triangle inequality for absolute values: \(|x + y| \leq |x| + |y|\) (https://mathworld.wolfram.com/TriangleInequality.html)

Thanks to 1, you won't have to prove that MAE is convex, but only that the absolute value function is convex.

If you start from the convexity definition, and you replace \(f(x)\) by \(|x|\), you should very quickly arrive at the triangle inequality equation (this helped me: \(\lambda|x| = |\lambda x|\) since lambda is positive)

PS: Sorry about the weird Latex...
[admin: fixed LaTeX. Use the inline/block LaTeX buttons in the editor.]

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