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Suppose I have a deep feed-forward neural network with sigmoid activation $\sigma$ already trained on a dataset $S$. Let's consider a training point $x_i \in S$. I want to analyze the entries of a hidden layer $h_{i,l}$, where

$$h_{i,l} = \sigma(W_l ( \sigma (W_{l-1} \sigma( \dots \sigma ( W_1 \cdot x_i))\dots). $$

My intuition would be that, since gradient descend has passed many times on the point $x_i$ updating the weights at every iteration, the entries of every hidden layer computed on $x_i$ would be either very close to zero or very close to one (thanks to the effect of the sigmoid activation).

Is this true? Is there a theoretical result in the literature which shows anything similar to this? Is there an empirical result which shows that?

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I do not have a specific example for you, but I think your intuition is wrong here. The loss function is not pushing activations of hidden layers to zeros or ones—unlike the activations of output layers.

Generally, if the activations of a sigmoid are close to zero or one (also called saturation), the gradients will be very small and the network will not learn—that is actually an undesirable property, sometimes called dead neurons, and there are many techniques such as batch normalization, which try to keep the activations of sigmoid units in their linear part.

To sum up, it may happen, but it is not a general rule, and it is something we try to avoid.

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