Convolutional Neural Networks:Step by Step

In this article, I will implement convolutional (CONV) and pooling (POOL) layers in numpy, including both forward propagation and backward propagation.

Before you read the text, I hope you remember the following symbols:

  • Superscript $[l]$ denotes an object of the $l^{th}$ layer.
    • Example: $a^{[4]}$ is the $4^{th}$ layer activation. $W^{[5]}$ and $b^{[5]}$ are the $5^{th}$ layer parameters.
  • Superscript $(i)$ denotes an object from the $i^{th}$ example.

    • Example: $x^{(i)}$ is the $i^{th}$ training example input.
  • Lowerscript $i$ denotes the $i^{th}$ entry of a vector.

    • Example: $a^{[l]}_i$ denotes the $i^{th}$ entry of the activations in layer $l$, assuming this is a fully connected (FC) layer.
  • $n_H$, $n_W$ and $n_C$ denote respectively the height, width and number of channels of a given layer. If you want to reference a specific layer $l$, you can also write $n_H^{[l]}$, $n_W^{[l]}$, $n_C^{[l]}$.

  • $n{H{prev}}$, $n{W{prev}}$ and $n{C{prev}}$ denote respectively the height, width and number of channels of the previous layer. If referencing a specific layer $l$, this could also be denoted $n_H^{[l-1]}$, $n_W^{[l-1]}$, $n_C^{[l-1]}$.

I assume that you are already familiar with numpy and/or have completed the previous article of this deep-learning theme articles. Let’s get started!

Outline of the Assignment

I will be implementing the building blocks of a convolutional neural network! Each function I will implement will have detailed instructions that will walk I through the steps needed:

  • Convolution functions, including:
    • Zero Padding
    • Convolve window
    • Convolution forward
    • Convolution backward
  • Pooling functions, including:
    • Pooling forward
    • Create mask
    • Distribute value
    • Pooling backward

This article will ask you to implement these functions from scratch in numpy. In the next article, I will use the TensorFlow equivalents of these functions to build the following model:

model

Note that for every forward function, there is its corresponding backward equivalent. Hence, at every step of your forward module you will store some parameters in a cache. These parameters are used to compute gradients during backpropagation.

Convolutional Neural Networks

Although programming frameworks make convolutions easy to use, they remain one of the hardest concepts to understand in Deep Learning. A convolution layer transforms an input volume into an output volume of different size, as shown below.

conv_nn

In this part, I will build every step of the convolution layer. I will first implement two helper functions: one for zero padding and the other for computing the convolution function itself.

Zero-Padding

Zero-padding adds zeros around the border of an image:

PAD

Figure 1 : Zero-Padding
Image (3 channels, RGB) with a padding of 2.

The main benefits of padding are the following:

  • It allows you to use a CONV layer without necessarily shrinking the height and width of the volumes. This is important for building deeper networks, since otherwise the height/width would shrink as you go to deeper layers. An important special case is the “same” convolution, in which the height/width is exactly preserved after one layer.

  • It helps us keep more of the information at the border of an image. Without padding, very few values at the next layer would be affected by pixels as the edges of an image.

What I implemented in the following function which pads all the images of a batch of examples X with zeros. Use np.pad. Note if we want to pad the array “a” of shape $(5,5,5,5,5)$ with pad = 1 for the 2nd dimension, pad = 3 for the 4th dimension and pad = 0 for the rest, we would do:

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a = np.pad(a, ((0,0), (1,1), (0,0), (3,3), (0,0)), 'constant', constant_values = (..,..))

Implement:

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def zero_pad(X, pad):
"""
Pad with zeros all images of the dataset X. The padding is applied to the height and width of an image,
as illustrated in Figure 1.

Argument:
X -- python numpy array of shape (m, n_H, n_W, n_C) representing a batch of m images
pad -- integer, amount of padding around each image on vertical and horizontal dimensions

Returns:
X_pad -- padded image of shape (m, n_H + 2*pad, n_W + 2*pad, n_C)
"""

X_pad = np.pad(X, ((0, 0), (pad, pad), (pad, pad), (0, 0)), mode='constant', constant_values=(0, 0))

return X_pad

Test code

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def demoZeroPad():
np.random.seed(1)
# (m, n_H, n_W, n_C)
x = np.random.randn(4, 3, 3, 2)
x_pad = zero_pad(x, 2)
print("x.shape =", x.shape)
print("x_pad.shape =", x_pad.shape)
print("x[1,1] =", x[1, 1])
print("x_pad[1,1] =", x_pad[1, 1])

fig, axarr = plt.subplots(1, 2)
axarr[0].set_title('x')
axarr[0].imshow(x[0, :, :, 0])
axarr[1].set_title('x_pad')
axarr[1].imshow(x_pad[0, :, :, 0])
plt.show()

Output:

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x.shape = (4, 3, 3, 2)
x_pad.shape = (4, 7, 7, 2)
x[1,1] = [[ 0.90085595 -0.68372786]
[-0.12289023 -0.93576943]
[-0.26788808 0.53035547]]
x_pad[1,1] = [[ 0. 0.]
[ 0. 0.]
[ 0. 0.]
[ 0. 0.]
[ 0. 0.]
[ 0. 0.]
[ 0. 0.]]

票

Single step of convolution

In this part,I will implement a single step of convolution, in which I apply the filter to a single position of the input. This will be used to build a convolutional unit, which:

  • Takes an input volume
  • Applies a filter at every position of the input
  • Outputs another volume (usually of different size)

Convolution_schematic

Figure 2 : Convolution operation
with a filter of 2x2 and a stride of 1 (stride = amount you move the window each time you slide)

In a computer vision application, each value in the matrix on the left corresponds to a single pixel value, and we convolve a 3x3 filter with the image by multiplying its values element-wise with the original matrix, then summing them up and adding a bias. In this first step of the exercise, I will implement a single step of convolution, corresponding to applying a filter to just one of the positions to get a single real-valued output.

Later in this article, I’ll apply this function to multiple positions of the input to implement the full convolutional operation.

conv_single_step(). Hint.

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def conv_single_step(a_slice_prev, W, b):
"""
Apply one filter defined by parameters W on a single slice (a_slice_prev) of the output activation
of the previous layer.

Arguments:
a_slice_prev -- slice of input data of shape (f, f, n_C_prev)
W -- Weight parameters contained in a window - matrix of shape (f, f, n_C_prev)
b -- Bias parameters contained in a window - matrix of shape (1, 1, 1)

Returns:
Z -- a scalar value, result of convolving the sliding window (W, b) on a slice x of the input data
"""

# Element-wise product between a_slice and W. Do not add the bias yet.
s = a_slice_prev * W
# Sum over all entries of the volume s.
Z = np.sum(s)
# Add bias b to Z. Cast b to a float() so that Z results in a scalar value.
Z = Z + float(b)

return Z

Convolutional Neural Networks - Forward pass

In the forward pass, we will take many filters and convolve them on the input. Each ‘convolution’ gives us a 2D matrix output. You will then stack these outputs to get a 3D volume:

Now I will implement the function below to convolve the filters W on an input activation A_prev. This function takes as input A_prev, the activations output by the previous layer (for a batch of m inputs), F filters/weights denoted by W, and a bias vector denoted by b, where each filter has its own (single) bias. Finally we also have access to the hyperparameters dictionary which contains the stride and the padding.

Hint:

  1. To select a 2x2 slice at the upper left corner of a matrix “a_prev” (shape (5,5,3)), I would do:
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    a_slice_prev = a_prev[0:2,0:2,:]

This will be useful when we will define a_slice_prev below, using the start/end indexes we will define.

  1. To define a_slice I will need to first define its corners vert_start, vert_end, horiz_start and horiz_end. This figure may be helpful for us to find how each of the corner can be defined using h, w, f and s in the code below.

    vert_horiz_kiank

Figure 3 : Definition of a slice using vertical and horizontal start/end (with a 2x2 filter)
This figure shows only a single channel.

Reminder:
The formulas relating the output shape of the convolution to the input shape is:

For this exercise, we won’t worry about vectorization, and will just implement everything with for-loops.

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def conv_forward(A_prev, W, b, hparameters):
"""
Implements the forward propagation for a convolution function

Arguments:
A_prev -- output activations of the previous layer, numpy array of shape (m, n_H_prev, n_W_prev, n_C_prev)
W -- Weights, numpy array of shape (f, f, n_C_prev, n_C)
b -- Biases, numpy array of shape (1, 1, 1, n_C)
hparameters -- python dictionary containing "stride" and "pad"

Returns:
Z -- conv output, numpy array of shape (m, n_H, n_W, n_C)
cache -- cache of values needed for the conv_backward() function
"""

# Retrieve dimensions from A_prev's shape (≈1 line)
(m, n_H_prev, n_W_prev, n_C_prev) = A_prev.shape

# Retrieve dimensions from W's shape (≈1 line)
(f, f, n_C_prev, n_C) = W.shape

# Retrieve information from "hparameters" (≈2 lines)
stride = hparameters['stride']
pad = hparameters['pad']

# Compute the dimensions of the CONV output volume using the formula given above. Hint: use int() to floor. (≈2 lines)
n_H = int((n_H_prev - f + 2 * pad) / stride) + 1
n_W = int((n_W_prev - f + 2 * pad) / stride) + 1

# Initialize the output volume Z with zeros. (≈1 line)
Z = np.zeros(shape=(m, n_H, n_W, n_C))

# Create A_prev_pad by padding A_prev
A_prev_pad = zero_pad(A_prev, pad)

for i in range(m): # loop over the batch of training examples
a_prev_pad = A_prev_pad[i] # Select ith training example's padded activation
for h in range(n_H): # loop over vertical axis of the output volume
for w in range(n_W): # loop over horizontal axis of the output volume
for c in range(n_C): # loop over channels (= #filters) of the output volume

# Find the corners of the current "slice" (≈4 lines)
vert_start = h * stride
vert_end = h * stride + f
horiz_start = w * stride
horiz_end = w * stride + f

# Use the corners to define the (3D) slice of a_prev_pad (See Hint above the cell). (≈1 line)
a_slice_prev = a_prev_pad[vert_start:vert_end, horiz_start:horiz_end, :]

# Convolve the (3D) slice with the correct filter W and bias b, to get back one output neuron. (≈1 line)
Z[i, h, w, c] = conv_single_step(a_slice_prev, W[:, :, :, c], b[:, :, :, c])

# Making sure your output shape is correct
assert (Z.shape == (m, n_H, n_W, n_C))

# Save information in "cache" for the backprop
cache = (A_prev, W, b, hparameters)

return Z, cache

Finally, CONV layer should also contain an activation, in which case we would add the following line of code:

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# Convolve the window to get back one output neuron
Z[i, h, w, c] = ...
# Apply activation
A[i, h, w, c] = activation(Z[i, h, w, c])

We don’t need to do it here.

Pooling layer

The pooling (POOL) layer reduces the height and width of the input. It helps reduce computation, as well as helps make feature detectors more invariant to its position in the input. The two types of pooling layers are:

  • Max-pooling layer: slides an ($f, f$) window over the input and stores the max value of the window in the output.
  • Average-pooling layer: slides an ($f, f$) window over the input and stores the average value of the window in the output.

These pooling layers have no parameters for backpropagation to train. However, they have hyperparameters such as the window size $f$. This specifies the height and width of the fxf window we would compute a max or average over.

Forward Pooling

Now, I am going to implement MAX-POOL and AVG-POOL, in the same function.

Reminder:
As there’s no padding, the formulas binding the output shape of the pooling to the input shape is:

Implement:

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def pool_forward(A_prev, hparameters, mode="max"):
"""
Implements the forward pass of the pooling layer

Arguments:
A_prev -- Input data, numpy array of shape (m, n_H_prev, n_W_prev, n_C_prev)
hparameters -- python dictionary containing "f" and "stride"
mode -- the pooling mode you would like to use, defined as a string ("max" or "average")

Returns:
A -- output of the pool layer, a numpy array of shape (m, n_H, n_W, n_C)
cache -- cache used in the backward pass of the pooling layer, contains the input and hparameters
"""

# Retrieve dimensions from the input shape
(m, n_H_prev, n_W_prev, n_C_prev) = A_prev.shape

# Retrieve hyperparameters from "hparameters"
f = hparameters["f"]
stride = hparameters["stride"]

# Define the dimensions of the output
n_H = int(1 + (n_H_prev - f) / stride)
n_W = int(1 + (n_W_prev - f) / stride)
n_C = n_C_prev

# Initialize output matrix A
A = np.zeros((m, n_H, n_W, n_C))

for i in range(m): # loop over the training examples
for h in range(n_H): # loop on the vertical axis of the output volume
for w in range(n_W): # loop on the horizontal axis of the output volume
for c in range(n_C): # loop over the channels of the output volume

# Find the corners of the current "slice" (≈4 lines)
vert_start = h * stride
vert_end = h * stride + f
horiz_start = w * stride
horiz_end = w * stride + f

# Use the corners to define the current slice on the ith training example of A_prev, channel c. (≈1 line)
a_prev_slice = A_prev[i, vert_start:vert_end, horiz_start:horiz_end, c]

# Compute the pooling operation on the slice. Use an if statment to differentiate the modes. Use np.max/np.mean.
if mode == "max":
A[i, h, w, c] = np.max(a_prev_slice)
elif mode == "average":
A[i, h, w, c] = np.average(a_prev_slice)


# Store the input and hparameters in "cache" for pool_backward()
cache = (A_prev, hparameters)

# Making sure your output shape is correct
assert (A.shape == (m, n_H, n_W, n_C))

return A, cache

We have now implemented the forward passes of all the layers of a convolutional network.

Backpropagation in convolutional neural networks

In modern deep learning frameworks, we only have to implement the forward pass, and the framework takes care of the backward pass, so most deep learning engineers don’t need to bother with the details of the backward pass. The backward pass for convolutional networks is complicated. If you wish however, you can work through this portion of the article to get a sense of what backprop in a convolutional network looks like.

When in an earlier course we implemented a simple (fully connected) neural network, we used backpropagation to compute the derivatives with respect to the cost to update the parameters. Similarly, in convolutional neural networks we can to calculate the derivatives with respect to the cost in order to update the parameters. The backprop equations are not trivial I briefly presented them below.

Convolutional layer backward pass

Let’s start by implementing the backward pass for a CONV layer.

Computing dA:

This is the formula for computing $dA$ with respect to the cost for a certain filter $W_c$ and a given training example:

Where $Wc$ is a filter and $dZ{hw}$ is a scalar corresponding to the gradient of the cost with respect to the output of the conv layer Z at the hth row and wth column (corresponding to the dot product taken at the ith stride left and jth stride down). Note that at each time, we multiply the the same filter $W_c$ by a different dZ when updating dA. We do so mainly because when computing the forward propagation, each filter is dotted and summed by a different a_slice. Therefore when computing the backprop for dA, we are just adding the gradients of all the a_slices.

In code, inside the appropriate for-loops, this formula translates into:

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da_prev_pad[vert_start:vert_end, horiz_start:horiz_end, :] += W[:,:,:,c] * dZ[i, h, w, c]

Computing dW:

This is the formula for computing $dW_c$ ($dW_c$ is the derivative of one filter) with respect to the loss:

Where $a{slice}$ corresponds to the slice which was used to generate the acitivation $Z{ij}$. Hence, this ends up giving us the gradient for $W$ with respect to that slice. Since it is the same $W$, we will just add up all such gradients to get $dW$.

In code, inside the appropriate for-loops, this formula translates into:

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dW[:,:,:,c] += a_slice * dZ[i, h, w, c]

Computing db:

This is the formula for computing $db$ with respect to the cost for a certain filter $W_c$:

As you have previously seen in basic neural networks, db is computed by summing $dZ$. In this case, we are just summing over all the gradients of the conv output (Z) with respect to the cost.

In code, inside the appropriate for-loops, this formula translates into:

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db[:,:,:,c] += dZ[i, h, w, c]

I will implement the conv_backward function below and sum over all the training examples, filters, heights, and widths. I will then compute the derivatives using formulas 1, 2 and 3 above.

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def conv_backward(dZ, cache):
"""
Implement the backward propagation for a convolution function

Arguments:
dZ -- gradient of the cost with respect to the output of the conv layer (Z), numpy array of shape (m, n_H, n_W, n_C)
cache -- cache of values needed for the conv_backward(), output of conv_forward()

Returns:
dA_prev -- gradient of the cost with respect to the input of the conv layer (A_prev),
numpy array of shape (m, n_H_prev, n_W_prev, n_C_prev)
dW -- gradient of the cost with respect to the weights of the conv layer (W)
numpy array of shape (f, f, n_C_prev, n_C)
db -- gradient of the cost with respect to the biases of the conv layer (b)
numpy array of shape (1, 1, 1, n_C)
"""

# Retrieve information from "cache"
(A_prev, W, b, hparameters) = cache

# Retrieve dimensions from A_prev's shape
(m, n_H_prev, n_W_prev, n_C_prev) = A_prev.shape

# Retrieve dimensions from W's shape
(f, f, n_C_prev, n_C) = W.shape

# Retrieve information from "hparameters"
stride = hparameters['stride']
pad = hparameters['pad']

# Retrieve dimensions from dZ's shape
(m, n_H, n_W, n_C) = dZ.shape

# Initialize dA_prev, dW, db with the correct shapes
dA_prev = np.zeros(shape=(m, n_H_prev, n_W_prev, n_C_prev))
dW = np.zeros(shape=(f, f, n_C_prev, n_C))
db = np.zeros(shape=(1, 1, 1, n_C))

# Pad A_prev and dA_prev
A_prev_pad = zero_pad(A_prev, pad)
dA_prev_pad = zero_pad(dA_prev, pad)

for i in range(m): # loop over the training examples

# select ith training example from A_prev_pad and dA_prev_pad
a_prev_pad = A_prev_pad[i]
da_prev_pad = dA_prev_pad[i]

for h in range(n_H): # loop over vertical axis of the output volume
for w in range(n_W): # loop over horizontal axis of the output volume
for c in range(n_C): # loop over the channels of the output volume

# Find the corners of the current "slice"
vert_start = h * stride
vert_end = h * stride + f
horiz_start = w * stride
horiz_end = w * stride + f

# Use the corners to define the slice from a_prev_pad
a_slice = a_prev_pad[vert_start:vert_end, horiz_start:horiz_end, :]

# Update gradients for the window and the filter's parameters using the code formulas given above
try:

da_prev_pad[vert_start:vert_end, horiz_start:horiz_end, :] += W[:, :, :, c] * dZ[i, h, w, c]
except Exception:
a = 0

dW[:, :, :, c] += a_slice * dZ[i, h, w, c]
db[:, :, :, c] += dZ[i, h, w, c]

# Set the ith training example's dA_prev to the unpaded da_prev_pad (Hint: use X[pad:-pad, pad:-pad, :])
dA_prev_pad[i, :, :, :] = da_prev_pad

# Making sure your output shape is correct
assert (dA_prev.shape == (m, n_H_prev, n_W_prev, n_C_prev))

return dA_prev, dW, db

Pooling layer - backward pass

Next, let’s implement the backward pass for the pooling layer, starting with the MAX-POOL layer. Even though a pooling layer has no parameters for backprop to update, we still need to backpropagation the gradient through the pooling layer in order to compute gradients for layers that came before the pooling layer.

Max pooling - backward pass

Before jumping into the backpropagation of the pooling layer, I am going to build a helper function called create_mask_from_window() which does the following:

As you can see, this function creates a “mask” matrix which keeps track of where the maximum of the matrix is. True (1) indicates the position of the maximum in X, the other entries are False (0). You’ll see later that the backward pass for average pooling will be similar to this but using a different mask.

I will implement create_mask_from_window() firstly. This function will be helpful for pooling backward.
Hints:

  • np.max() may be helpful. It computes the maximum of an array.
  • If we have a matrix X and a scalar x: A = (X == x) will return a matrix A of the same size as X such that:

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    A[i,j] = True if X[i,j] = x
    A[i,j] = False if X[i,j] != x
  • Here, we don’t need to consider cases where there are several maxima in a matrix.

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def create_mask_from_window(x):
"""
Creates a mask from an input matrix x, to identify the max entry of x.

Arguments:
x -- Array of shape (f, f)

Returns:
mask -- Array of the same shape as window, contains a True at the position corresponding to the max entry of x.
"""

mask = x == np.max(x)

return mask

Why do we keep track of the position of the max? It’s because this is the input value that ultimately influenced the output, and therefore the cost. Backprop is computing gradients with respect to the cost, so anything that influences the ultimate cost should have a non-zero gradient. So, backprop will “propagate” the gradient back to this particular input value that had influenced the cost.

Average pooling - backward pass

In max pooling, for each input window, all the “influence” on the output came from a single input value—the max. In average pooling, every element of the input window has equal influence on the output. So to implement backprop, I will now implement a helper function that reflects this.

For example if we did average pooling in the forward pass using a 2x2 filter, then the mask you’ll use for the backward pass will look like:

This implies that each position in the $dZ$ matrix contributes equally to output because in the forward pass, we took an average.

I will implement the function below to equally distribute a value dz through a matrix of dimension shape. Hint

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def distribute_value(dz, shape):
"""
Distributes the input value in the matrix of dimension shape

Arguments:
dz -- input scalar
shape -- the shape (n_H, n_W) of the output matrix for which we want to distribute the value of dz

Returns:
a -- Array of size (n_H, n_W) for which we distributed the value of dz
"""

# Retrieve dimensions from shape (≈1 line)
(n_H, n_W) = shape

# Compute the value to distribute on the matrix (≈1 line)
average = dz / (n_W * n_H)

# Create a matrix where every entry is the "average" value (≈1 line)
a = np.ones(shape=shape) * average

return a

Putting it together: Pooling backward

We now have everything we need to compute backward propagation on a pooling layer.

Now I will implement the pool_backward function in both modes ("max" and "average"). I will once again use 4 for-loops (iterating over training examples, height, width, and channels). I will use an if/elif statement to see if the mode is equal to 'max' or 'average'. If it is equal to ‘average’ I will use the distribute_value() function I implemented above to create a matrix of the same shape as a_slice. Otherwise, the mode is equal to ‘max‘, and I will create a mask with create_mask_from_window() and multiply it by the corresponding value of dZ.

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def pool_backward(dA, cache, mode="max"):
"""
Implements the backward pass of the pooling layer

Arguments:
dA -- gradient of cost with respect to the output of the pooling layer, same shape as A
cache -- cache output from the forward pass of the pooling layer, contains the layer's input and hparameters
mode -- the pooling mode you would like to use, defined as a string ("max" or "average")

Returns:
dA_prev -- gradient of cost with respect to the input of the pooling layer, same shape as A_prev
"""


# Retrieve information from cache (≈1 line)
(A_prev, hparameters) = cache

# Retrieve hyperparameters from "hparameters" (≈2 lines)
stride = hparameters['stride']
f = hparameters['f']

# Retrieve dimensions from A_prev's shape and dA's shape (≈2 lines)
m, n_H_prev, n_W_prev, n_C_prev = A_prev.shape
m, n_H, n_W, n_C = dA.shape

# Initialize dA_prev with zeros (≈1 line)
dA_prev = np.zeros(shape=(m, n_H_prev, n_W_prev, n_C_prev))

for i in range(m): # loop over the training examples

# select training example from A_prev (≈1 line)
a_prev = A_prev[i]

for h in range(n_H): # loop on the vertical axis
for w in range(n_W): # loop on the horizontal axis
for c in range(n_C): # loop over the channels (depth)

# Find the corners of the current "slice" (≈4 lines)
vert_start = h * stride
vert_end = h * stride + f
horiz_start = w * stride
horiz_end = w * stride + f

# Compute the backward propagation in both modes.
if mode == "max":

# Use the corners and "c" to define the current slice from a_prev (≈1 line)
a_prev_slice = a_prev[vert_start:vert_end, horiz_start:horiz_end, c]
# Create the mask from a_prev_slice (≈1 line)
mask = create_mask_from_window(a_prev_slice)
# Set dA_prev to be dA_prev + (the mask multiplied by the correct entry of dA) (≈1 line)
dA_prev[i, vert_start: vert_end, horiz_start: horiz_end, c] += mask * dA[i, h, w, c]

elif mode == "average":

# Get the value a from dA (≈1 line)
da = dA[i, h, w, c]
# Define the shape of the filter as fxf (≈1 line)
shape = (f, f)
# Distribute it to get the correct slice of dA_prev. i.e. Add the distributed value of da. (≈1 line)
dA_prev[i, vert_start: vert_end, horiz_start: horiz_end, c] += distribute_value(da, shape)


# Making sure your output shape is correct
assert (dA_prev.shape == A_prev.shape)

return dA_prev

Congratulations

Congratulation on completing this article. You now understand how convolutional neural networks work. You have implemented all the building blocks of a neural network. In the next article I will implement a ConvNet using TensorFlow.

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