Convolutional Neural Networks: Step by Step
Welcome to Course 4's first assignment! In this assignment, you will implement convolutional (CONV) and pooling (POOL) layers in numpy, including both forward propagation and (optionally) backward propagation.
Notation:
- Superscript [l] denotes an object of the lth layer.
- Example: a[4] is the 4th layer activation. W[5] and b[5] are the 5th layer parameters.
- Superscript (i) denotes an object from the ith example.
- Example: x(i) is the ith training example input.
- Lowerscript i denotes the ith entry of a vector.
- Example: a[l]i denotes the ith entry of the activations in layer l, assuming this is a fully connected (FC) layer.
- nH, nW and nC 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[l]H, n[l]W, n[l]C.
- nHprev, nWprev and nCprev 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[l−1]H, n[l−1]W, n[l−1]C.
We assume that you are already familiar with numpy and/or have completed the previous courses of the specialization. Let's get started!
1 - Packages
Let's first import all the packages that you will need during this assignment.
- numpy is the fundamental package for scientific computing with Python.
- matplotlib is a library to plot graphs in Python.
- np.random.seed(1) is used to keep all the random function calls consistent. It will help us grade your work.
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import numpy as np import h5py import matplotlib.pyplot as plt %matplotlib inline plt.rcParams['figure.figsize'] = (5.0, 4.0) # set default size of plots plt.rcParams['image.interpolation'] = 'nearest' plt.rcParams['image.cmap'] = 'gray' %load_ext autoreload %autoreload 2 np.random.seed(1)2 - Outline of the Assignment
You will be implementing the building blocks of a convolutional neural network! Each function you will implement will have detailed instructions that will walk you through the steps needed:
- Convolution functions, including:
- Zero Padding
- Convolve window
- Convolution forward
- Convolution backward (optional)
- Pooling functions, including:
- Pooling forward
- Create mask
- Distribute value
- Pooling backward (optional)
-
This notebook will ask you to implement these functions from scratch in
numpy. In the next notebook, you will use the TensorFlow equivalents of these functions to build the following model: -
-
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# GRADED FUNCTION: zero_pad 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) """ ### START CODE HERE ### (≈ 1 line) X_pad = np.pad(X, ((0,0), (pad,pad), (pad,pad),(0,0)), 'constant') ### END CODE HERE ### return X_pad np.random.seed(1) 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])
-
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# GRADED FUNCTION: conv_single_step 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 """ ### START CODE HERE ### (≈ 2 lines of code) # 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) ### END CODE HERE ### return Z np.random.seed(1) a_slice_prev = np.random.randn(4, 4, 3) W = np.random.randn(4, 4, 3) b = np.random.randn(1, 1, 1) Z = conv_single_step(a_slice_prev, W, b) print("Z =", Z)Z = -6.99908945068 
# GRADED FUNCTION: conv_forward
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
"""
### START CODE HERE ###
# 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+2*pad-f)/stride+1)
n_W = int((n_W_prev+2*pad-f)/stride+1)
# Initialize the output volume Z with zeros. (≈1 line)
Z = np.zeros((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 = vert_start+f
horiz_start = w*stride
horiz_end = horiz_start+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])
### END CODE HERE ###
# 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
np.random.seed(1)
A_prev = np.random.randn(10,4,4,3)
W = np.random.randn(2,2,3,8)
b = np.random.randn(1,1,1,8)
hparameters = {"pad" : 2,
"stride": 2}
Z, cache_conv = conv_forward(A_prev, W, b, hparameters)
print("Z's mean =", np.mean(Z))
print("Z[3,2,1] =", Z[3,2,1])
print("cache_conv[0][1][2][3] =", cache_conv[0][1][2][3])Z's mean = 0.0489952035289 Z[3,2,1] = [-0.61490741 -6.7439236 -2.55153897 1.75698377 3.56208902 0.53036437 5.18531798 8.75898442] cache_conv[0][1][2][3] = [-0.20075807 0.18656139 0.41005165]
Expected Output:
| Z's mean | 0.0489952035289 |
| Z[3,2,1] | [-0.61490741 -6.7439236 -2.55153897 1.75698377 3.56208902 0.53036437 5.18531798 8.75898442] |
| cache_conv[0][1][2][3] | [-0.20075807 0.18656139 0.41005165] |
Finally, CONV layer should also contain an activation, in which case we would add the following line of code:
# 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])
You don't need to do it here.
4 - 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:
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Max-pooling layer: slides an (f,f) window over the input and stores the max value of the window in the output.
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Average-pooling layer: slides an (f,f) window over the input and stores the average value of the window in the output.
# GRADED FUNCTION: pool_forward
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))
### START CODE HERE ###
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.mean(a_prev_slice)
### END CODE HERE ###
# 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
np.random.seed(1)
A_prev = np.random.randn(2, 4, 4, 3)
hparameters = {"stride" : 2, "f": 3}
A, cache = pool_forward(A_prev, hparameters)
print("mode = max")
print("A =", A)
print()
A, cache = pool_forward(A_prev, hparameters, mode = "average")
print("mode = average")
print("A =", A)
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)
"""
### START CODE HERE ###
# 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((A_prev.shape))
dW = np.zeros((W.shape))
db = np.zeros((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
da_prev_pad[vert_start:vert_end, horiz_start:horiz_end, :] +=
W[:,:,:,c] * dZ[i, h, w, c]
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[i, :, :, :] =da_prev_pad[pad:-pad, pad:-pad, :]
### END CODE HERE ###
# 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, dbnp.random.seed(1)
dA, dW, db = conv_backward(Z, cache_conv)
print("dA_mean =", np.mean(dA))
print("dW_mean =", np.mean(dW))
print("db_mean =", np.mean(db))
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.
"""
### START CODE HERE ### (≈1 line)
mask = (x==np.max(x))
### END CODE HERE ###
return mask
np.random.seed(1)
x = np.random.randn(2,3)
mask = create_mask_from_window(x)
print('x = ', x)
print("mask = ", mask)x = [[ 1.62434536 -0.61175641 -0.52817175] [-1.07296862 0.86540763 -2.3015387 ]] mask = [[ True False False] [False False False]]
Expected Output:
| x = | [[ 1.62434536 -0.61175641 -0.52817175] [-1.07296862 0.86540763 -2.3015387 ]] |
| mask = | [[ True False False] [False False False]] |
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.
5.2.2 - 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, you 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:dZ=1→dZ=[1/41/4
1/41/4]
This implies that each position in the dZ matrix contributes equally to output because in the forward pass, we took an average.
Exercise: Implement the function below to equally distribute a value dz through a matrix of dimension shape. Hint
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
"""
### START CODE HERE ###
# Retrieve dimensions from shape (≈1 line)
(n_H, n_W) = shape
# Compute the value to distribute on the matrix (≈1 line)
average =dz/(n_H*n_W)
# Create a matrix where every entry is the "average" value (≈1 line)
a = np.zeros((n_H, n_W))+average
### END CODE HERE ###
return a
a = distribute_value(2, (2,2))
print('distributed value =', a)distributed value = [[ 0.5 0.5] [ 0.5 0.5]]
5.2.3 Putting it together: Pooling backward
You now have everything you need to compute backward propagation on a pooling layer.
Exercise: Implement the pool_backward function in both modes ("max" and "average"). You will once again use 4 for-loops (iterating over training examples, height, width, and channels). You should use an if/elif statement to see if the mode is equal to 'max' or 'average'. If it is equal to 'average' you should use the distribute_value() function you implemented above to create a matrix of the same shape as a_slice. Otherwise, the mode is equal to 'max', and you will create a mask with create_mask_from_window() and multiply it by the corresponding value of dZ.
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
"""
### START CODE HERE ###
# 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((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 = vert_start+f
horiz_start = w*stride
horiz_end = horiz_start+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)
### END CODE ###
# Making sure your output shape is correct
assert(dA_prev.shape == A_prev.shape)
return dA_prevnp.random.seed(1)
A_prev = np.random.randn(5, 5, 3, 2)
hparameters = {"stride" : 1, "f": 2}
A, cache = pool_forward(A_prev, hparameters)
dA = np.random.randn(5, 4, 2, 2)
dA_prev = pool_backward(dA, cache, mode = "max")
print("mode = max")
print('mean of dA = ', np.mean(dA))
print('dA_prev[1,1] = ', dA_prev[1,1])
print()
dA_prev = pool_backward(dA, cache, mode = "average")
print("mode = average")
print('mean of dA = ', np.mean(dA))
print('dA_prev[1,1] = ', dA_prev[1,1]) 