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][l] denotes an object of the lthlth layer.
  • Example: a[4]a[4] is the 4th4th layer activation. W[5]W[5] and b[5]b[5] are the 5th5th layer parameters.

• Superscript (i)(i) denotes an object from the ithith example.
  • Example: x(i)x(i) is the ithith training example input.

• Lowerscript ii denotes the ithith entry of a vector.
  • Example: a[l]iai[l] denotes the ithith entry of the activations in layer ll, assuming this is a fully connected (FC) layer.

• nHnH, nWnW and nCnC denote respectively the height, width and number of channels of a given layer. If you want to reference a specific layer ll, you can also write n[l]HnH[l], n[l]WnW[l], n[l]CnC[l].
• nHprevnHprev, nWprevnWprev and nCprevnCprev denote respectively the height, width and number of channels of the previous layer. If referencing a specific layer ll, this could also be denoted n[l?1]HnH[l?1], n[l?1]WnW[l?1], n[l?1]CnC[l?1].

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.

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:

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.

3 - 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.

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

3.1 - Zero-Padding

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

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.

Exercise: Implement the following function, which pads all the images of a batch of examples X with zeros. Use np.pad. Note if you want to pad the array "a" of shape (5,5,5,5,5)(5,5,5,5,5) with pad = 1 for the 2nd dimension, pad = 3 for the 4th dimension and pad = 0 for the rest, you would do:

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

Expected Output:

3.2 - Single step of convolution

In this part, implement a single step of convolution, in which you 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)

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, you 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 notebook, you‘ll apply this function to multiple positions of the input to implement the full convolutional operation.

Exercise: Implement conv_single_step(). Hint.

Expected Output:

3.3 - Convolutional Neural Networks - Forward pass

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

Exercise: 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 you 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)), you would do:

   a_slice_prev = a_prev[0:2,0:2,:]
   

   This will be useful when you will define a_slice_prev below, using the start/end indexes you will define.
2. To define a_slice you will need to first define its corners vert_start, vert_end, horiz_start and horiz_end. This figure may be helpful for you to find how each of the corner can be defined using h, w, f and s in the code below.

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.

Expected Output:

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:

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

• Average-pooling layer: slides an (f,ff,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 ff. This specifies the height and width of the fxf window you would compute a max or average over.

4.1 - Forward Pooling

Now, you are going to implement MAX-POOL and AVG-POOL, in the same function.

Exercise: Implement the forward pass of the pooling layer. Follow the hints in the comments below.

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

Expected Output:

Congratulations! You have now implemented the forward passes of all the layers of a convolutional network.

The remainer of this notebook is optional, and will not be graded.

5 - Backpropagation in convolutional neural networks (OPTIONAL / UNGRADED)

In modern deep learning frameworks, you 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 optional portion of the notebook to get a sense of what backprop in a convolutional network looks like.

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

5.1 - Convolutional layer backward pass

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

5.1.1 - Computing dA:

This is the formula for computing dAdA with respect to the cost for a certain filter WcWc and a given training example:

Where WcWc is a filter and dZhwdZhw 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 WcWc 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:

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

5.1.2 - Computing dW:

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

Where asliceaslice corresponds to the slice which was used to generate the acitivation ZijZij. Hence, this ends up giving us the gradient for WW with respect to that slice. Since it is the same WW, we will just add up all such gradients to get dWdW.

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

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

5.1.3 - Computing db:

This is the formula for computing dbdb with respect to the cost for a certain filter WcWc:

As you have previously seen in basic neural networks, db is computed by summing dZdZ. In this case, you 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:

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

Exercise: Implement the conv_backward function below. You should sum over all the training examples, filters, heights, and widths. You should then compute the derivatives using formulas 1, 2 and 3 above.

Expected Output:

5.2 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, you still need to backpropagation the gradient through the pooling layer in order to compute gradients for layers that came before the pooling layer.

5.2.1 Max pooling - backward pass

Before jumping into the backpropagation of the pooling layer, you are 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.

Exercise: Implement create_mask_from_window(). This function will be helpful for pooling backward. Hints:

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

  A[i,j] = True if X[i,j] = x
  A[i,j] = False if X[i,j] != x
  

• Here, you don‘t need to consider cases where there are several maxima in a matrix.

Expected Output:

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:

This implies that each position in the dZdZ 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

Expected Output:

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.

Expected Output:

mode = max:

mode = average

Congratulations !

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