# Why do activation functions have to be monotonic?

I am currently preparing for an exam on neural networks. In several protocols from former exams I read that the activation functions of neurons (in multilayer perceptrons) have to be monotonic.

I understand that activation functions should be differentiable, have a derivative which is not 0 on most points, and be non-linear. I do not understand why being monotonic is important/helpful.

I know the following activation functions and that they are monotonic:

• ReLU
• Sigmoid
• Tanh
• Softmax: I'm not sure if the definition of monotonicity is applicable for functions $$f: \mathbb{R}^n \rightarrow \mathbb{R}^m$$ with $$n, m > 1$$
• Softplus
• (Identity)

However, I still can't see any reason why for example $$\varphi(x) = x^2$$.

Why do activation functions have to be monotonic?

(Related side question: is there any reason why the logarithm/exponential function is not used as an activation function?)

• – Franck Dernoncourt Dec 7 '15 at 1:13
• @MartinThoma Are you sure softmax is monotonic? – Media Feb 21 '18 at 7:07
• Thanks @Media. To answer your question: I'm not sure what "monotonic" even means for functions in $f:R^n \rightarrow R^m$ with $m > 1$. For $m=1$ softmax is constant and thus monotonic. But without defining $<$ for elements in $R^n$ with $n>1$ I don't think monotonic makes any sense. – Martin Thoma Feb 21 '18 at 19:50
• @MartinThoma Thanks, actually it was also a question of mine. I didn't know, and still don't know, if there is an extension for monotonic in functions with multiple outputs. Math stuff, you know! – Media Feb 22 '18 at 14:06