Activation function after pooling layer or convolutional layer?

Question

The theory from these links show that the order of Convolutional Network is: Convolutional Layer - Non-linear Activation - Pooling Layer.

1. Neural networks and deep learning (equation (125)
2. Deep learning book (page 304, 1st paragraph)
3. Lenet (the equation)
4. The source in this headline

But, in the last implementation from those sites, it said that the order is: Convolutional Layer - Pooling Layer - Non-linear Activation

1. network3.py
2. The sourcecode, LeNetConvPoolLayer class

I've tried too to explore a Conv2D operation syntax, but there is no activation function, it's only convolution with flipped kernel. Can someone help me to explain why is this happen?

Answer 1

Well, max-pooling and monotonely increasing non-linearities commute. This means that MaxPool(Relu(x)) = Relu(MaxPool(x)) for any input. So the result is the same in that case. So it is technically better to first subsample through max-pooling and then apply the non-linearity (if it is costly, such as the sigmoid). In practice it is often done the other way round - it doesn't seem to change much in performance.

As for conv2D, it does not flip the kernel. It implements exactly the definition of convolution. This is a linear operation, so you have to add the non-linearity yourself in the next step, e.g. theano.tensor.nnet.relu.

Answer 2

In many papers people use conv -> pooling -> non-linearity. It does not mean that you can't use another order and get reasonable results. In case of max-pooling layer and ReLU the order does not matter (both calculate the same thing):

You can proof that this is the case by remembering that ReLU is an element-wise operation and a non-decreasing function so

The same thing happens for almost every activation function (most of them are non-decreasing). But does not work for a general pooling layer (average-pooling).

---

Nonetheless both orders produce the same result, Activation(MaxPool(x)) does it significantly faster by doing less amount of operations. For a pooling layer of size k, it uses k^2 times less calls to activation function.

Sadly this optimization is negligible for CNN, because majority of the time is used in convolutional layers.