<p>The first answer is from 2015 and a bit of age.</p>
<p>Today, CNNs typically also use batchnorm - while there is some debate why this helps: the inventors mention covariate shift: <a href="https://arxiv.org/abs/1502.03167" rel="nofollow noreferrer">https://arxiv.org/abs/1502.03167</a>
There are other theories like smoothing the loss landscape: <a href="https://arxiv.org/abs/1805.11604" rel="nofollow noreferrer">https://arxiv.org/abs/1805.11604</a></p>
<p>Either way, it is a method that helps to deal significantly with vanishing/exploding gradient problem that is also relevant for CNNs. In CNNs you also apply the chain rule to get gradients. That is the update of the first layer is proportional to the product of N numbers, where N is the number of inputs. It is very likely that this number is either relatively big or small compared to the update of the last layer. This might be seen by looking at the variance of a product of random variables that quickly grows the more variables are being multiplied: <a href="https://stats.stackexchange.com/questions/52646/variance-of-product-of-multiple-random-variables">https://stats.stackexchange.com/questions/52646/variance-of-product-of-multiple-random-variables</a></p>
<p>For recurrent networks that have long sequences of inputs, ie. of length L, the situation is often worse than for CNN, since there the product consists of L numbers. Often the sequence length L in a RNN is much larger than the number of layers N in a CNN.</p>
Do convolutional neural networks suffer from the vanishing gradient?
<p>I think I read somewhere that convolutional neural networks do not suffer from the vanishing gradient problem as much as standard sigmoid neural networks with increasing number of layers. But I have not been able to find a 'why'.</p>

<p>Does it truly not suffer from the problem or am I wrong and it depends on the activation function?
[I have been using Rectified Linear Units, so I have never tested the Sigmoid Units for Convolutional Neural Networks]</p>