---

title: ‘DeepLearning.ai作业:(4-1)-- 卷积神经网络（Foundations of CNN）’
id: dl-ai-4-1h
tags:

• dl.ai
• homework
  categories:
• AI
• Deep Learning
  date: 2018-09-30 16:07:23

---

首发于个人博客:fangzh.top，欢迎来访
本周的作业分为了两部分：

• 卷积神经网络的模型搭建
• 用TensorFlow来训练卷积神经网络

Part1：Convolutional Neural Networks: Step by Step

主要内容：

• convolution funtions:
  • Zero Padding
  • Convolve window
  • Convolution forward
  • Convolution backward (optional)
• Pooling functions：
  • Pooling forward
  • Create mask
  • Distribute value
  • Pooling backward (optional)

Convolutional Neural Networks

创建CNN的主要函数

1. Zero Padding

先创建一个padding函数，用来输入图像X，输出padding后的图像，这里使用的是np.pad()函数，

a = np.pad(a, ((0,0), (1,1), (0,0), (3,3), (0,0)), 'constant', constant_values = (..,..))
表示a有5个维度，在第1维的两边都填上1个pad，和第3维的两边都填上3个pad，constant_values表示两边要填充的值

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', constant_values=(0,0))
    ### END CODE HERE ###
    
    return X_pad

2.Single step of convolution

创建一个单步的卷积运算，也就是一次输入一个切片，大小和卷积核相同，对应元素相乘再求和，最后再加个bias项。

# 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

3.Convolutional Neural Networks - Forward pass

创建一次完整的卷积过程，也就是利用上面的一次卷积，进行for循环。进行切片的时候，注意边界vert_start, vert_end, horiz_start and horiz_end

这一步应该先弄清楚A_prev，A，W，b的维度，超参数项包括了stride和pad

$$n_H=\lfloor \frac{n_{H_{prev}}-f+2\times pad}{stride}\rfloor +1$$
$$n_W=\lfloor \frac{n_{W_{prev}}-f+2\times pad}{stride}\rfloor +1$$
$$n_C=\text{number of filters used in the convolution}$$

# 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 = 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])
                                        
    ### 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

Pooling layer

创建池化层，注意得到的维度需要向下取整，用int()对float()进行转换

$$n_H=\lfloor \frac{n_{H_{prev}}-f}{stride}\rfloor +1$$
$$n_W=\lfloor \frac{n_{W_{prev}}-f}{stride}\rfloor +1$$
$$n_C=n_{C_{prev}}$$

同样需要先进行切边，而后分为max和average两种，分别用np.max和np.mean

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 = vert_start + f
                    horiz_start = w * stride
                    horiz_end = horiz_start + 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

Backpropagation in convolutional neural networks

卷积神经网络的求导是比较难以理解的，这里有卷积层的求导和池化层的求导。

1.Convolutional layer backward pass

假设经过卷积层后我们的输出$Z=W\times A+b$

那么反向传播过程中需要求的就是$dA,dW,db$，其中$dA$是原输入的数据，包含了原图像中的每一个像素，

而这个时候假设从后面传过来的$dZ$是已经知道的。

1.计算dA

从公式可以看出，$dA=W\times dZ$，具体一点，$dA$的每一个切片就是$W_c$乘上$dZ$在输出图片的每一个像素的求和结果，从矩阵的角度，每一次$W_c\times d{Z}_{hw}$得到的就是从**单个输出的图片像素到输入图片切片（大小为W）**的映射。因此公式为：

$$dA+=\sum _{h=0}^{n_H}\sum _{w=0}^{n_W}{W}_c\times d{Z}_{hw}$$

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

2.计算dW

$dW=A\times dZ$，而更具体一点，因为W对Z的每一个像素都是有作用的，所以就等于每一个输入图片的切片乘以对应输出图片像素的导数，然后再求和！

$$d{W}_c+=\sum _{h=0}^{n_H}\sum _{w=0}^{n_W}{a}_{slice}\times d{Z}_{hw}$$

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

3.计算db

$$db=\sum _h\sum _{w}d{Z}_{hw}$$

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

所以得到以下：

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(b.shape)

    # 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, db

Pooling layer - backward pass

这里max pooling和average poolling要分开处理。

1. Max pooling - backward pass

假设pool size是$2\times 2$的，那么，4个像素中只有1个留下来了，其余的都没有效果了，所以在max pooling中，从后面传递过来的导数值，只作用在max的那个元素，而且继续往前传递，不做任何改动，在其余3个元素的导数都是0。

创建一个mask矩阵，让最大值为1，其余的都为0，这样子就可以作为一个映射矩阵向前映射了。

KaTeX parse error: Expected & or \\ or \cr or \end at position 107: …trix} 0 && 0 \\\̲ ̲1 && 0 \end{bma…

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

2. Average pooling - backward pass

和max不同，average pooling相当于把backward传过来的值分成了$n_H\times n_W$等分。所以要计算的参数就比max pooling多很多了，这也就是为什么一般都用max pooling，不用average pooling

KaTeX parse error: Expected & or \\ or \cr or \end at position 67: …} 1/4 && 1/4 \\\̲ ̲1/4 && 1/4 \end…

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 = n_H * n_W
    
    # Create a matrix where every entry is the "average" value (≈1 line)
    a = dz / average * np.ones((n_H, n_W))
    ### END CODE HERE ###
    
    return a

结合两种方法：

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(A_prev.shape)
    
    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_prev

Part2：Convolutional Neural Networks: Application

用TensorFlow来搭建卷积神经网络。

1.Create placeholders

先创建placeholders，用来训练中传递X,Y

def create_placeholders(n_H0, n_W0, n_C0, n_y):
    """
    Creates the placeholders for the tensorflow session.
    
    Arguments:
    n_H0 -- scalar, height of an input image
    n_W0 -- scalar, width of an input image
    n_C0 -- scalar, number of channels of the input
    n_y -- scalar, number of classes
        
    Returns:
    X -- placeholder for the data input, of shape [None, n_H0, n_W0, n_C0] and dtype "float"
    Y -- placeholder for the input labels, of shape [None, n_y] and dtype "float"
    """

    ### START CODE HERE ### (≈2 lines)
    X = tf.placeholder(tf.float32, shape=(None,n_H0, n_W0, n_C0))
    Y = tf.placeholder(tf.float32, shape=(None,n_y))
    ### END CODE HERE ###
    
    return X, Y

2.Initialize parameters

用来初始化参数，主要是W1,W2,在这里就没有用b了

用W = tf.get_variable("W", [1,2,3,4], initializer = ...)

initializer 用tf.contrib.layers.xavier_initializer

# GRADED FUNCTION: initialize_parameters

def initialize_parameters():
    """
    Initializes weight parameters to build a neural network with tensorflow. The shapes are:
                        W1 : [4, 4, 3, 8]
                        W2 : [2, 2, 8, 16]
    Returns:
    parameters -- a dictionary of tensors containing W1, W2
    """
    
    tf.set_random_seed(1)                              # so that your "random" numbers match ours
        
    ### START CODE HERE ### (approx. 2 lines of code)
    W1 = tf.get_variable('W1', [4, 4, 3, 8],initializer= tf.contrib.layers.xavier_initializer(seed = 0 ))
    W2 = tf.get_variable('W2', [2, 2, 8, 16],initializer= tf.contrib.layers.xavier_initializer(seed = 0))
    ### END CODE HERE ###

    parameters = {"W1": W1,
                  "W2": W2}
    
    return parameters

记得这只是创建了图而已，并没有真正的初始化参数，在执行中还需要

init = tf.global_variables_initializer()

sess_test.run(init)

3. Forward propagation

模型为：CONV2D -> RELU -> MAXPOOL -> CONV2D -> RELU -> MAXPOOL -> FLATTEN -> FULLYCONNECTED

 - Conv2D: stride 1, padding is "SAME"
 - ReLU
 - Max pool: Use an 8 by 8 filter size and an 8 by 8 stride, padding is "SAME"
 - Conv2D: stride 1, padding is "SAME"
 - ReLU
 - Max pool: Use a 4 by 4 filter size and a 4 by 4 stride, padding is "SAME"
 - Flatten the previous output.
 - FULLYCONNECTED (FC) layer：这里全连接层不需要有激活函数，因为后面计算cost的时候会加上softmax，因此这里不需要加

用到的函数：

• tf.nn.conv2d(X,W1, strides = [1,s,s,1], padding = ‘SAME’): given an input $X$ and a group of filters $W1$, this function convolves $W1$'s filters on X. The third input ([1,f,f,1]) represents the strides for each dimension of the input (m, n_H_prev, n_W_prev, n_C_prev). You can read the full documentation here

• tf.nn.max_pool(A, ksize = [1,f,f,1], strides = [1,s,s,1], padding = ‘SAME’): given an input A, this function uses a window of size (f, f) and strides of size (s, s) to carry out max pooling over each window. You can read the full documentation here

• tf.nn.relu(Z1): computes the elementwise ReLU of Z1 (which can be any shape). You can read the full documentation here.

• tf.contrib.layers.flatten§: given an input P, this function flattens each example into a 1D vector it while maintaining the batch-size. It returns a flattened tensor with shape [batch_size, k]. You can read the full documentation here.

• tf.contrib.layers.fully_connected(F, num_outputs): given a the flattened input F, it returns the output computed using a fully connected layer. You can read the full documentation here.

In the last function above (tf.contrib.layers.fully_connected), the fully connected layer automatically initializes weights in the graph and keeps on training them as you train the model. Hence, you did not need to initialize those weights when initializing the parameters.

# GRADED FUNCTION: forward_propagation

def forward_propagation(X, parameters):
    """
    Implements the forward propagation for the model:
    CONV2D -> RELU -> MAXPOOL -> CONV2D -> RELU -> MAXPOOL -> FLATTEN -> FULLYCONNECTED
    
    Arguments:
    X -- input dataset placeholder, of shape (input size, number of examples)
    parameters -- python dictionary containing your parameters "W1", "W2"
                  the shapes are given in initialize_parameters

    Returns:
    Z3 -- the output of the last LINEAR unit
    """
    
    # Retrieve the parameters from the dictionary "parameters" 
    W1 = parameters['W1']
    W2 = parameters['W2']
    
    ### START CODE HERE ###
    # CONV2D: stride of 1, padding 'SAME'
    Z1 = tf.nn.conv2d(X, filter=W1, strides=[1,1,1,1],padding='SAME')
    # RELU
    A1 = tf.nn.relu(Z1)
    # MAXPOOL: window 8x8, sride 8, padding 'SAME'
    P1 = tf.nn.max_pool(A1,ksize=[1, 8, 8, 1], strides=[1, 8, 8, 1],padding='SAME')
    # CONV2D: filters W2, stride 1, padding 'SAME'
    Z2 = tf.nn.conv2d(P1, filter=W2, strides=[1, 1, 1, 1],padding='SAME')
    # RELU
    A2 = tf.nn.relu(Z2)
    # MAXPOOL: window 4x4, stride 4, padding 'SAME'
    P2 = tf.nn.max_pool(A2,ksize=[1, 4, 4, 1], strides=[1, 4, 4, 1],padding='SAME')
    # FLATTEN
    P2 = tf.contrib.layers.flatten(P2)
    # FULLY-CONNECTED without non-linear activation function (not not call softmax).
    # 6 neurons in output layer. Hint: one of the arguments should be "activation_fn=None" 
    Z3 = tf.contrib.layers.fully_connected(P2, 6,activation_fn=None)
    ### END CODE HERE ###

    return Z3

4. Compute cost

• tf.nn.softmax_cross_entropy_with_logits(logits = Z3, labels = Y): computes the softmax entropy loss. This function both computes the softmax activation function as well as the resulting loss. You can check the full documentation here.这个函数已经包含了计算softmax，还有求cross-entropy两件事了。
• tf.reduce_mean: computes the mean of elements across dimensions of a tensor. Use this to sum the losses over all the examples to get the overall cost. You can check the full documentation here.

# GRADED FUNCTION: compute_cost 

def compute_cost(Z3, Y):
    """
    Computes the cost
    
    Arguments:
    Z3 -- output of forward propagation (output of the last LINEAR unit), of shape (6, number of examples)
    Y -- "true" labels vector placeholder, same shape as Z3
    
    Returns:
    cost - Tensor of the cost function
    """
    
    ### START CODE HERE ### (1 line of code)
    cost = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(logits=Z3,labels=Y))
    ### END CODE HERE ###
    
    return cost

5. Model

把前面的函数都结合起来，创建一个完整的模型。

其中random_mini_batches()已经给我们了，优化器使用了

optimizer = tf.train.AdamOptimizer(learning_rate).minimize(cost)

# GRADED FUNCTION: model

def model(X_train, Y_train, X_test, Y_test, learning_rate = 0.009,
          num_epochs = 100, minibatch_size = 64, print_cost = True):
    """
    Implements a three-layer ConvNet in Tensorflow:
    CONV2D -> RELU -> MAXPOOL -> CONV2D -> RELU -> MAXPOOL -> FLATTEN -> FULLYCONNECTED
    
    Arguments:
    X_train -- training set, of shape (None, 64, 64, 3)
    Y_train -- test set, of shape (None, n_y = 6)
    X_test -- training set, of shape (None, 64, 64, 3)
    Y_test -- test set, of shape (None, n_y = 6)
    learning_rate -- learning rate of the optimization
    num_epochs -- number of epochs of the optimization loop
    minibatch_size -- size of a minibatch
    print_cost -- True to print the cost every 100 epochs
    
    Returns:
    train_accuracy -- real number, accuracy on the train set (X_train)
    test_accuracy -- real number, testing accuracy on the test set (X_test)
    parameters -- parameters learnt by the model. They can then be used to predict.
    """
    
    ops.reset_default_graph()                         # to be able to rerun the model without overwriting tf variables
    tf.set_random_seed(1)                             # to keep results consistent (tensorflow seed)
    seed = 3                                          # to keep results consistent (numpy seed)
    (m, n_H0, n_W0, n_C0) = X_train.shape             
    n_y = Y_train.shape[1]                            
    costs = []                                        # To keep track of the cost
    
    # Create Placeholders of the correct shape
    ### START CODE HERE ### (1 line)
    X, Y = create_placeholders(n_H0, n_W0,n_C0,n_y)
    ### END CODE HERE ###

    # Initialize parameters
    ### START CODE HERE ### (1 line)
    parameters = initialize_parameters()
    ### END CODE HERE ###
    
    # Forward propagation: Build the forward propagation in the tensorflow graph
    ### START CODE HERE ### (1 line)
    Z3 = forward_propagation(X,parameters)
    ### END CODE HERE ###
    
    # Cost function: Add cost function to tensorflow graph
    ### START CODE HERE ### (1 line)
    cost = compute_cost(Z3, Y)
    ### END CODE HERE ###
    
    # Backpropagation: Define the tensorflow optimizer. Use an AdamOptimizer that minimizes the cost.
    ### START CODE HERE ### (1 line)
    optimizer = tf.train.AdamOptimizer(learning_rate).minimize(cost)
    ### END CODE HERE ###
    
    # Initialize all the variables globally
    init = tf.global_variables_initializer()
     
    # Start the session to compute the tensorflow graph
    with tf.Session() as sess:
        
        # Run the initialization
        sess.run(init)
        
        # Do the training loop
        for epoch in range(num_epochs):

            minibatch_cost = 0.
            num_minibatches = int(m / minibatch_size) # number of minibatches of size minibatch_size in the train set
            seed = seed + 1
            minibatches = random_mini_batches(X_train, Y_train, minibatch_size, seed)

            for minibatch in minibatches:

                # Select a minibatch
                (minibatch_X, minibatch_Y) = minibatch
                # IMPORTANT: The line that runs the graph on a minibatch.
                # Run the session to execute the optimizer and the cost, the feedict should contain a minibatch for (X,Y).
                ### START CODE HERE ### (1 line)
                _ , temp_cost = sess.run([optimizer,cost],feed_dict={X:minibatch_X,Y:minibatch_Y})
                ### END CODE HERE ###
                
                minibatch_cost += temp_cost / num_minibatches
                

            # Print the cost every epoch
            if print_cost == True and epoch % 5 == 0:
                print ("Cost after epoch %i: %f" % (epoch, minibatch_cost))
            if print_cost == True and epoch % 1 == 0:
                costs.append(minibatch_cost)
        
        
        # plot the cost
        plt.plot(np.squeeze(costs))
        plt.ylabel('cost')
        plt.xlabel('iterations (per tens)')
        plt.title("Learning rate =" + str(learning_rate))
        plt.show()

        # Calculate the correct predictions
        predict_op = tf.argmax(Z3, 1)
        correct_prediction = tf.equal(predict_op, tf.argmax(Y, 1))
        
        # Calculate accuracy on the test set
        accuracy = tf.reduce_mean(tf.cast(correct_prediction, "float"))
        print(accuracy)
        train_accuracy = accuracy.eval({X: X_train, Y: Y_train})
        test_accuracy = accuracy.eval({X: X_test, Y: Y_test})
        print("Train Accuracy:", train_accuracy)
        print("Test Accuracy:", test_accuracy)
                
        return train_accuracy, test_accuracy, parameters

得到效果如图：