Program CNN

Convolutional Neural Networks

Zero-Padding

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 = (..,..))
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)), 'constant', constant_values=0)

    return X_pad

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)

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.

Reminder: The formulas relating the output shape of the convolution to the input shape is: nH=⌊nHprev−f+2×padstride⌋+1n_H = \lfloor \frac{n_{H_{prev}} - f + 2 \times pad}{stride} \rfloor +1 nW=⌊nWprev−f+2×padstride⌋+1n_W = \lfloor \frac{n_{W_{prev}} - f + 2 \times pad}{stride} \rfloor +1 nC=number of filters used in the convolutionn_C = \text{number of filters used in the convolution}

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

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

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 you would compute a max or average over.

Forward Pooling

Reminder: As there's no padding, the formulas binding the output shape of the pooling to the input shape is: nH=⌊nHprev−fstride⌋+1n_H = \lfloor \frac{n_{H_{prev}} - f}{stride} \rfloor +1 nW=⌊nWprev−fstride⌋+1n_W = \lfloor \frac{n_{W_{prev}} - f}{stride} \rfloor +1 nC=nCprevn_C = n_{C_{prev}}

Backpropagation in convolutional neural networks

Convolutional layer backward pass

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:

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:

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, 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:

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.

注意：卷积层

A_prev：(m, n_H_prev, n_W_prev, n_C_prev)

W：(f, f, n_C_prev, n_C)， b：(1, 1, 1, n_C)

输出Z：(m, n_H, n_W, n_C) 其中 n_H = 1 + int((n_H_prev + 2 pad - f) / stride)，n_W = 1 + int((n_W_prev + 2 pad - f) / stride)

经过激活函数后A：(m, n_H, n_W, n_C) （这里没有）

Pooling layer - backward pass

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.

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:

  • Here, you don't need to consider cases where there are several maxima in a matrix.

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, 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 $dZ$ 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

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.

注意：池化层

A_prev：(m, n_H_prev, n_W_prev, n_C_prev)

输出A：(m, n_H, n_W, n_C) 其中 n_H = int(1 + (n_H_prev - f) / stride)，n_W = int(1 + (n_W_prev - f) / stride)，n_C = n_C_prev

TensorFlow

TensorFlow model

Create placeholders

Implement the function below to create placeholders for the input image X and the output Y. You should not define the number of training examples for the moment. To do so, you could use "None" as the batch size, it will give you the flexibility to choose it later. Hence X should be of dimension [None, n_H0, n_W0, n_C0] and Y should be of dimension [None, n_y]. Hint.

Initialize parameters

You will initialize weights/filters $W_1$ and $W_2$ using tf.contrib.layers.xavier_initializer(seed = 0). You don't need to worry about bias variables as you will soon see that TensorFlow functions take care of the bias. Note also that you will only initialize the weights/filters for the conv2d functions. TensorFlow initializes the layers for the fully connected part automatically.

Reminder - to initialize a parameter W of shape [1,2,3,4] in Tensorflow, use:

More Info.

Forward propagation

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