Example of convex activation function

Question

For a particular task, I need a convex activation function with the following properties:

• f''(x) > 0
• 0 <= f(x) <= 1
• f(x) is monotonic
• f(x) is not "exploding" i.e. avoiding functions such as f(x) = x²

The only example I have in mind for this is the softplus activation function. Would you have anything else in mind?

Answer 1

There is no such function.

A convex function defined on all of $\mathbb{R}$ with bounded range is constant.

Softplus is not bounded above as required by your second bullet.