D5.5 · LESSON 5 OF 6

Pooling

Calculate a convolutional output and explain filters, stride, padding, pooling and residual connections.

THE IDEA

Pooling summarises a neighbourhood

Max pooling slides a window (usually 2×2, stride 2) and keeps only the largest value in each. The map gets smaller (4×4 becomes 2×2), later layers do less work, and small shifts of a pattern inside a window change nothing. Average pooling keeps the mean instead. Global average pooling averages each whole channel into a single number, which is a common way to end a convolutional network.

bright dot at (x=1, y=1)

after 2×2 max pooling

Pooling keeps the strongest response in each window, so the dot can move inside its window without changing the output. It is not completely shift-proof: crossing a window boundary changes it.

Move the bright dot. Inside a window the pooled output stays the same; across a boundary it changes.

Predict first. What is max pooling over the window [1, 4, 2, 3]?

Chapter context

A convolutional layer reuses a local filter across positions. Weight sharing encodes a structural assumption about repeated local patterns while greatly reducing parameters compared with a fully connected map over all pixels.

TRY IT YOURSELF

Part 1. Do it hands-on

Pool a grid by hand, then test what shifting does. The Python part writes max, average and global pooling.

bright dot at (x=1, y=1)

after 2×2 max pooling

Pooling keeps the strongest response in each window, so the dot can move inside its window without changing the output. It is not completely shift-proof: crossing a window boundary changes it.

Max-pool this 4×4 grid (2×2 windows, stride 2):

results in reading order:

To do: Max-pool the 4×4 grid with a 2×2 window and stride 2. Type the four results in reading order (top-left, top-right, bottom-left, bottom-right)
To do: A bright dot starts at (1, 1). Move it one pixel so the pooled 2×2 output does NOT change
To do: Now move it one pixel so the pooled output DOES change

Part 2. Do it in code

max_pool2d(x, k, s=None): x is a 2-D array. Take the maximum of every k×k window, moving by s (default s = k). Windows must fit entirely inside x (so leftover rows and columns are ignored). Return a 2-D array.
avg_pool2d(x, k, s=None): the same with the mean.
global_avg_pool(x): x has shape (C, H, W). Return a vector of length C with the mean of each channel.

import numpy as np

def max_pool2d(x, k, s=None):
    pass

def avg_pool2d(x, k, s=None):
    pass

def global_avg_pool(x):
    pass

Ctrl/⌘ + Enter runs it

Stuck, or something not working? You can come back to it any time.

CHECK YOUR UNDERSTANDING

What is max pooling over [1,4,2,3]?

AFour, because max pooling keeps the largest value in the window.BTen, because pooling adds up the values in the window.CTwo and a half, because pooling averages the window.

Chapter reference: math, worked example & code

This section belongs to the whole chapter, so it is the same on every lesson in it and may cover a different lesson's topic. The code for the experiment you just ran is inside the experiment itself.

Math and notation

For one spatial dimension, output size is

n_{out}=\left\lfloor\frac{n+2p-d(k-1)-1}{s}\right\rfloor+1.

Here n is input size, k kernel size, p padding, d dilation and s stride. Cross-correlation is $y_t=\sum_iw_i x_{t+i}$ . A residual block has form $h_{out}=h+F(h)$ when shapes agree.

Worked example

For input [1,2,3,4] and kernel [1,−1], stride-one valid cross-correlation returns [−1,−1,−1]. Separately, n=7,k=3,p=1,s=2,d=1 gives floor((7+2−2−1)/2)+1=4 output positions.

Reference code

The following example uses the Python standard library and does not call a remote service.

x,w = [1,2,3,4],[1,-1]
y = [sum(x[t+i]*w[i] for i in range(len(w))) for t in range(len(x)-len(w)+1)]
assert y == [-1,-1,-1]
n,k,p,s,d = 7,3,1,2,1
assert (n+2*p-d*(k-1)-1)//s+1 == 4

Python examples are provided for study and local execution; this page does not execute Python.

Common mistakes

Padding does not reveal missing real pixels. Kernel flipping conventions change calculations. A residual shortcut needs a defined shape-compatible path.