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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?

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There is no such function.

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

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