課程四(Convolutional Neural Networks)，第一周（Foundations of Convolutional Neural Networks） —— 2.Programming assignments：Convolutional Model: step by step

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 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_i^[l]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_Hprev, n_Wprev and n_Cprev 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].

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.

【code】

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:

 

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.

【中文翻譯】

2-任務的概要

您將實現卷積神經網絡的構建塊!您將實現的每個功能將有詳細的說明, 將指導您完成所需的步驟:

卷積函數, 包括:

　　填充0值

　　卷積窗口

　　卷積前向傳播

　　卷積后向傳播(可選)

池化函數, 包括:

　　池化 前向傳播

　　創建掩碼

　　分布值

　　池化后向傳播(可選)

本筆記本將要求您在 numpy 中從頭開始實現這些函數。在下一本筆記本中, 您將使用這些函數的 TensorFlow 等效項來構建以下模型:

請注意, 對於每個前向傳播, 都有相應的后向傳播。因此, 在前向傳播模塊中的每個步驟中, 您都將在緩存中存儲一些參數。這些參數用於在反向傳播期間計算導數。

 

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:

                                             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.

【中文翻譯】 

填充的主要優點如下:

　　它允許您使用一個卷積層，不需要縮小圖片高度和寬度。這對於建立更深的網絡很重要, 因為到網絡更深的層, 高度/寬度就會縮小。一個重要特殊情況是 "same" 卷積, 高度或寬度在每一層以后確切地被保留。

　　它可以幫助我們保留圖像的邊框上更多的信息。如果沒有填充, 那么下一層的值受到圖片邊緣像素的影響將很少。

 

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)(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 = (..,..))

 

【code】

 

# 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', constant_values=(0,0) )
    ### 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])          #編號為1的樣本，編號為1的通道
print ("x_pad[1,1] =", x_pad[1,1])
# print ("x=",x) 此處可以打印x,查看x

fig, axarr = plt.subplots(1, 2) # 1行2個窗口
axarr[0].imshow(x[0,:,:,0])  #序號為0，通道為0的樣本
axarr[1].set_title('x_pad')
axarr[1].imshow(x_pad[0,:,:,0])  #序號為0，通道為0的樣本

 

【result】

 

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.]]

 

<matplotlib.image.AxesImage at 0x7f94350437f0>

 

 

Expected Output:

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.]]

 

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)

                         Figure2: 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, 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.

【code】

 

# 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 + np.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)

 

【result】

Z = -6.99908945068

 

 

Expected Output:

Z	-6.99908945068

 

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:

 

                                                   注：原網頁是是視頻，此處為截圖

 

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.

                    

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.

 

【中文翻譯】

練習：實現下面的函數, 在輸入激活值 A_prev 上 卷積 過濾器 W。該函數的輸入有 ：A_prev、前一層的激活值輸出 (對於一批 m樣本的 輸入)、F 過濾器/權重 (W 表示) 和由 b 表示的偏置向量, 其中每個過濾器都有自己的 (單) 偏置。最后, 您還可以訪問包含步幅（stride）和填充(padding)的參數字典。

 

提示：

1.要選擇矩陣 "a_prev" (形狀 (5，5，3)) 左上角的2x2 切片, 請執行以下操作:

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

 

當您在下面定義 a_slice_prev 時, 使用將定義的開始/結束索引, 這將非常有用。

 

2.要定義 a_slice, 您需要首先定義其各個角： vert_start、vert_end、horiz_start 和 horiz_end。 此圖可能有助於您在下面的代碼中，如何使用 h、w、f 和 s 來定義每個角。

圖 3: 使用垂直和水平 開始/結束 (帶有2x2 的過濾器) 的切片定義

此圖僅顯示單個通道。

 

提示: 卷積的輸出形狀與輸入形狀之間關系的公式為:

 

對於這個練習, 我們不會擔心矢量化, 只會用 for 循環實現一切。

 

【code】

 

# 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)  從 A_prev 的形狀中檢索維度 
    (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)/pad +1)
    n_W =int((n_W_prev +  2*pad - f)/pad +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 =  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]) #需要找到對應的濾波器，所以 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

 

【result】

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.

 

-------------------------------------------------------------------------------------

【整理者注釋：對以下代碼塊的解釋，詳見插圖】

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

-------------------------------------------------------------------------------------

 

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,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.

【中文翻譯】

池 化(POOL) 層減少了輸入的高度和寬度。它有助於減少計算, 也有助於使特征檢測器在輸入中的位置變化的更少。有兩種類型的池化層: 

　　最大池層: 在輸入上滑動一個 (f，f) 窗口, 並在輸出中存儲窗口的最大值。

　　平均池層: 在輸入上滑動一個 (f、f) 窗口, 並將窗口的平均值存儲在輸出中。

 

 

這些池層沒有用於訓練的反向傳播的參數。然而, 他們有超參數，如窗口大小 f。這將指定要計算最大值或平均值的 fxf 窗口的高度和寬度。

 

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:

【code】

# 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 =  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] #此處要對每一個通道取窗口，注意與卷積區別
                    
                    # 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)  

# print(A.shape) (2,1,1,3)

 

【result】

mode = max
A = [[[[ 1.74481176  0.86540763  1.13376944]]]
     [[[ 1.13162939  1.51981682  2.18557541]]]]

mode = average
A = [[[[ 0.02105773 -0.20328806 -0.40389855]]]
     [[[-0.22154621  0.51716526  0.48155844]]]]

Expected Output:

A =	[[[[ 1.74481176 0.86540763 1.13376944]]] [[[ 1.13162939 1.51981682 2.18557541]]]]
A =	[[[[ 0.02105773 -0.20328806 -0.40389855]]] [[[-0.22154621 0.51716526 0.48155844]]]]

 

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.

 

 【中文翻譯】

在現代的深層學習框架中, 您只需要實現前向傳播, 而框架負責向后傳播, 因此大多數深學習的工程師不必費心處理向后傳播的細節。卷積網絡的向后傳播是復雜的。但是, 如果你希望, 你可以通過這個可選部分的筆記, 以了解什么 反向傳播 在一個卷積網絡是如何實現的。

 

在早期的課程中, 您實現了一個簡單的 (完全連接的) 神經網絡, 您使用反向傳播來計算與更新成本函數對參數的導數。同樣, 在卷積神經網絡中, 為了更新參數, 您可以計算與成本函數有關的導數。反向傳播方程不是微不足道的, 我們沒有在講課中推導出來, 但我們在下面簡要介紹了它們。

 

 

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 dA with respect to the cost for a certain filter W_c and a given training example:

 

Where W_c 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:

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 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:

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

 

5.1.3 - 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:

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

 

 

【中文翻譯】

5.1-卷積層向后傳播

讓我們從實現一個卷積層的向后傳播開始。

 

5.1.1-計算 dA:

這是對於一個給定的訓練示例、過濾器W_c ，通過成本函數計算dA的公式：

其中 W_c 是一個過濾器, dZ_hw 是一個標量，是成本函數對卷積層輸出Z的第h行第w列的值的導數。(對應在第 i 步向左和 第j步向下的點成)。請注意, 每次更新 dA 時, 我們都會用不同的 dZ 乘以相同的過濾器W_c。我們這樣做主要是因為在計算正向傳播時, 每個過濾器都是用不同的 a_slice 進行點乘和求和的。因此, 當反向傳播幾算dA時, 我們把所有對 a_slices 的梯度加起來。 

在代碼中, 在適當的 for 循環內, 此公式轉換為:

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

 

5.1.2-計算 dW_c:

 這是計算 dW_c(dW_c 是一個成本函數對過濾器的導數) 的公式:

 

 a_slice 對應於用於生成激活值 Z_ij 的切片。因此, 這最終給了我們該切片 的W 的梯度。因為它是相同的 W, 我們將所有這樣的梯度相加, 以獲得 dW。

在代碼中, 在適當的 for 循環內, 此公式轉換為:

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

 

5.1.3 - 計算 db:

這是計算 db 的公式（成本函數對某一過濾器W_c ）:

正如您以前在基本神經網絡中看到的, db 是通過 dZ求和 計算的。在這種情況下, 您只是在對成本函數的所有輸出 (Z) 的導數進行求和。

在代碼中, 在適當的 for 循環內, 此公式轉換為:

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.

 【code】

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.random.randn(m, n_H_prev, n_W_prev, n_C_prev)                          
    dW = np.random.randn(f, f, n_C_prev, n_C)
    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 =  vert_start + f
                    horiz_start = w * stride
                    horiz_end =  horiz_start + 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[i,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

 

np.random.seed(1)
dA, dW, db = conv_backward(Z, cache_conv) #此處傳入的為什么是Z,而不是dZ  ???
print("dA_mean =", np.mean(dA))
print("dW_mean =", np.mean(dW))
print("db_mean =", np.mean(db))

【result】

 

dA_mean = 1.50871418576
dW_mean = 1.80043206846
db_mean = 7.83923256462

Expected Output:

dA_mean    1.45243777754
dW_mean    1.72699145831
db_mean    7.83923256462

 

-------------------------------------------------------------------------------------

【整理者注釋：解釋以下代碼】

 # 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[i,pad:-pad, pad:-pad, :]

 

【問題】：為什么  dA_prev[i, :, :, :] =dA_prev_pad[i,pad:-pad, pad:-pad, :] ？

【用代碼驗證如下】

# (n_H,n_W,n_C)=(1,1,2)

a=np.array([[[1,2]]])   #臨時造的數據
print("a.shape=",a.shape)  

a.shape= (1, 1, 2)

 

# 進行pad操作，pad=1, 則(n_H_pad,n_W_pad,n_C_pad)=(3,3,2)

b=np.array([[[1,2],[3,4],[3,4]],[[7,8],[1,2],[3,4]],[[7,8],[1,2],[3,4]]]) #臨時造的數據
print("b.shape=",b.shape) 

b.shape= (3, 3, 2)

 

# 驗證 b[pad:-pad, pad:-pad, :] 的shape是否與a的shape一致
pad=1
c=b[pad:-pad, pad:-pad, :]
print("c.shape=",c.shape) 

assert(a.shape==c.shape)
#運行結果沒有報錯，所以 b[pad:-pad, pad:-pad, :] 的shape是否與a的shape一致
#從運行結果上也可以看出，shape都為（1,1,2）

 

c.shape= (1, 1, 2)

print("a=",a)
print("c=",c)     #a和c是一樣的

a= [[[1 2]]]
c= [[[1 2]]]

-------------------------------------------------------------------------------------

 

 

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.

【code】

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)

 

【result】

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:

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

【code】

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.ones( (n_H, n_W))*average
    ### END CODE HERE ###
    
    return a

 

a = distribute_value(2, (2,2))
print('distributed value =', a)

 

【result】

distributed value = [[ 0.5  0.5]
                     [ 0.5  0.5]]

 

 

Expected Output:

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.

【code】

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

np.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]) 

 

【result】

mode = max
mean of dA =  0.145713902729
dA_prev[1,1] =  [[ 0.          0.        ]
 [ 5.05844394 -1.68282702]
 [ 0.          0.        ]]

mode = average
mean of dA =  0.145713902729
dA_prev[1,1] =  [[ 0.08485462  0.2787552 ]
 [ 1.26461098 -0.25749373]
 [ 1.17975636 -0.53624893]]

Expected Output:

mode = max:

mean of dA =    0.145713902729
dA_prev[1,1] =    [[ 0. 0. ] 
[ 5.05844394 -1.68282702] 
[ 0. 0. ]]
mode = average

mean of dA =    0.145713902729
dA_prev[1,1] =    [[ 0.08485462 0.2787552 ] 
[ 1.26461098 -0.25749373] 
[ 1.17975636 -0.53624893]]

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.