# What is the linear region of the sigmoid function?

In lecture 9.1 of Geoffrey Hinton's course Neural Networks for Machine Learning, he offers a justification for why "early stopping" as a method to improve generalisation works.

He states that using a network with logistic hidden units and small weights, the total inputs to these hidden units will be close to 0 and therefore in the middle of the linear range of the sigmoid and will thus behave like a linear network (having a lower capacity / expressive power).

Where is the linear region of the sigmoid function? I would have thought that it would be in the middle of the curve, around 0 on the horizontal axis. (I've highlighted the area in question in the image below)

However, since he mentions that the total inputs being close to 0 when the sigmoid behaves in a linear fashion it sounds like he's referring to the curved taper to 0 as the output (in the bottom left of the picture).

Where is the linear region of the sigmoid?

• He's probably referring to the curve 2*sigmoid(t) - 1, which scales and re-centers to that the curve passes through the origin and asymptotes to minus one and one. Commented May 3, 2018 at 20:48

Any function can be approximated with a linear function with certain "accuracy", how accurate do you want and how do you quantify the accuracy you want? These two questions are the key to answer what is the region of $x$.