Gfyuki

This is a small conceptual question that's been nagging me for a while: How can we back-propagate through a max-pooling layer in a neural network?

I came across max-pooling layers while going through this tutorial for Torch 7's nn library. The library abstracts the gradient calculation and forward passes for each layer of a deep network. I don't understand how the gradient calculation is done for a max-pooling layer.

I know that if you have an input ${z_i}^l$ going into neuron $i$ of layer $l$, then ${delta_i}^l$ (defined as ${delta_i}^l = frac{partial E}{partial {z_i}^l}$) is given by:
$$
{delta_i}^l = theta^{'}({z_i}^l) sum_{j} {delta_j}^{l+1} w_{i,j}^{l,l+1}
$$

So, a max-pooling layer would receive the ${delta_j}^{l+1}$'s of the next layer as usual; but since the activation function for the max-pooling neurons takes in a vector of values (over which it maxes) as input, ${delta_i}^{l}$ isn't a single number anymore, but a vector ($theta^{'}({z_j}^l)$ would have to be replaced by $nabla theta(left{{z_j}^lright})$). Furthermore, $theta$, being the max function, isn't differentiable with respect to it's inputs.

So....how should it work out exactly?

There is no gradient with respect to non maximum values, since changing them slightly does not affect the output. Further the max is locally linear with slope 1, with respect to the input that actually achieves the max. Thus, the gradient from the next layer is passed back to only that neuron which achieved the max. All other neurons get zero gradient.

So in your example, $delta_i^l$ would be a vector of all zeros, except that the $i^*$th location will get a values $left{delta_j^{l+1}right}$ where $i^* = argmax_{i} (z_i^l)$

Max Pooling

So suppose you have a layer P which comes on top of a layer PR. Then the forward pass will be something like this:

$ P_i = f(sum_j W_{ij} PR_j)$,

where $P_i$ is the activation of the ith neuron of the layer P, f is the activation function and W are the weights. So if you derive that, by the chain rule you get that the gradients flow as follows:

$grad(PR_j) = sum_i grad(P_i) f^prime W_{ij}$.

But now, if you have max pooling, $f = id$ for the max neuron and $f = 0$ for all other neurons, so $f^prime = 1$ for the max neuron in the previous layer and $f^prime = 0$ for all other neurons. So:

$grad(PR_{max neuron}) = sum_i grad(P_i) W_{i {max neuron}}$,

$grad(PR_{others}) = 0.$

@Shinvu's answer is well written, I would like to point to a video that explains the gradient of Max() operation well within a computational graph which is quick to grasp.!