In training a neural network, you often see the curve showing how fast the neural network is converging (or not)getting better. It usually grows very fast then slows down to almost horizontal.
Is there a mathematical formula that matches these curves?
Some similar curves are:
$$y=1-e^{-x}$$
$$y=\frac{x}{1+x}$$
$$y=\tanh(x)$$
$$y=1+x-\sqrt{1+x^2}$$
Is there a theoretical reason for this shape?