# What does the term saturating nonlinearities mean?

I was reading the paper ImageNet Classification with Deep Convolutional Neural Networks and in section 3 were they explain the architecture of their Convolutional Neural Network they explain how they preferred using:

non-saturating nonlinearity $f(x) = max(0, x).$

because it was faster to train. In that paper they seem to refer to saturating nonlinearities as the more traditional functions used in CNNs, the sigmoid and the hyperbolic tanget functions (i.e. $f(x) = tanh(x)$ and $f(x) = \frac{1}{1 + e^{-x}} = (1 + e^{-x})^{-1}$ as saturating). However, I was wondering why they refer to these functions as "saturating" or "non-saturating"? In what sense are these function "saturating" or "non-saturating"? What do those terms mean in the context of convolutional neural networks? Are they used in other areas of machine learning (and statistics)?

Basically, what do they mean by "saturating" or "non-saturating" non-linearities?

## Intuition

A saturating activation function squeezes the input.

## Definitions

• $f$ is non-saturating iff $(|\lim_{z\to-\infty} f(z)| = +\infty) \vee |\lim_{z\to+\infty} f(z)| = +\infty)$
• $f$ is saturating iff $f$ is not non-saturating.

These definitions are not specific to convolutional neural networks.

## Examples

The Rectified Linear Unit (ReLU) activation function, which is defined as $f(x)=max(0,x)$ is non-saturating because $\lim_{z\to+\infty} f(z) = +\infty$:

The sigmoid activation function, which is defined as $f(x) = \frac{1}{1 + e^{-x}}$ is saturating, because it squashes real numbers to range between $[0,1]$:

The tanh activation function is saturating as it squashes real numbers to range between $[-1,1]$:

(figures are from CS231n, MIT License)

• ah, nice makes sense! I know this wasn't my original question, but what is that property important in the context of ML and CNNs? – Charlie Parker Sep 28 '15 at 15:43
• For ANNs, to avoid having one unit with a large output that impacts too much the ANN's output layer. – Franck Dernoncourt Sep 28 '15 at 18:03
• whats the difference between tan and sigmoid? both of them squash the numbers in a closed range! I dont get it, Can you elaborate this abit more? I'm kind of bad in mathematics . (by the way I'm coming from a CNN perspective ) – Breeze Feb 17 '16 at 11:09
• @FranckDernoncourt Did you mean saturating for tanh activation function? I guess there is a typo? :) – CoderSpinoza Mar 24 '16 at 4:44
• Can someone tell me why they chose this word? Is there an intuition behind that? Is it an import from a foreign language or another field? – quietContest Apr 8 '17 at 5:06

The most common activation functions are LOG and TanH. These functions have a compact range, meaning that they compress the neural response into a bounded subset of the real numbers. The LOG compresses inputs to outputs between 0 and 1, the TAN H between -1 and 1. These functions display limiting behavior at the boundaries.

At the border the gradient of the output with respect to the input ∂yj/∂xj is very small. So Gradient is small hence small steps to convergence hence longer time to converge.