# 2D max pool gradient propagation

I am trying to understand gradient propagation for a 2D max pooling operation when there is multiple filters for each position in the 2D grid (i.e. size = $$b\times2\times2\times d$$, where $$b$$ is batch size and $$d$$ is the number of filters).

From my understanding, max values get given the input gradients, whereas all non-max values get a gradient of 0 as a change of their input does not affect the result.

So as an example say $$b=1$$ and $$d=2$$, I perform a max pool with kernel size $$[2, 2]$$. On the forward pass:

$$\begin{bmatrix} [1, 2] & [4,6] \\ [3,7] & [2,2]\end{bmatrix}$$ $$\to$$ $$[[4, 7]]$$ (max of first and second dimensions respectively).

Say on backpropagation the input gradients $$g=[1, 2]$$, would the output gradients of the max pool operation be:

$$\begin{bmatrix} [0, 0] & [1,0] \\ [0,2] & [0,0]\end{bmatrix}$$

I have only seen examples where $$d=1$$ so wanted to make sure I am doing this correctly.

## 1 Answer

Your understanding is correct. For the sake of clarity, the max pooling operation can be understood as a rectifier function defined by (see Wikipedia entry for further details), $$f(x) = max(0, x)$$ whose derivative is $$f^{'}(x) = 1 \text{ if } y \geq 0, 0 \text{ otherwise}$$ where we have chosen the derivative to be 1 at 0. But here, instead of comparing against 0, you compare against the maximum value within the pooling mask.