# Backpropagation Through Max-Pooling Layer

## Introduction

I have once come up with a question “how do we do back propagation through max-pooling layer?”. The short answer is “there is no gradient with respect to non-maximum values”.

## Proof

Max-pooling is defined as

$$

y = \max(x_1, x_2, \cdots, x_n)

$$

where $y$ is the output and $x_i$ is the value of the neuron.

Alternatively, we could consider max-pooling layer as an affine layer without bias terms. The weight matrix in this affine layer is not trainable though.

Concretely, for the output $y$ after max-pooling, we have

$$

y = \sum_{i=1}^{n} w_i x_i

$$

where

$$

w_i =

\begin{cases}

1 & \text{if } x_i = \max(x_1, x_2, \cdots, x_n) \\

0 & \text{otherwise}

\end{cases}

$$

The gradient to each neuron is

$$

\begin{aligned}

\frac{\partial y}{\partial x_i} &= w_i \\

&=

\begin{cases}

1 & \text{if } x_i = \max(x_1, x_2, \cdots, x_n) \\

0 & \text{otherwise}

\end{cases}

\end{aligned}

$$

This simply means that the gradient to the neuron is 1 for the neuron with the maximum value. The gradients for all the other neurons are 0. The neurons whose gradients are 0 do not contribute to the gradients in the earlier neurons due to the chain rule.

## Conclusions

There is no gradient with respect to non-maximum values.

Backpropagation Through Max-Pooling Layer