DeepLearning.ai深度学习课程笔记
  • Introduction
  • 第一门课 神经网络和深度学习(Neural-Networks-and-Deep-Learning)
    • 第一周：深度学习引言(Introduction to Deep Learning)
      • 1.1 神经网络的监督学习(Supervised Learning with Neural Networks)
      • 1.2 为什么神经网络会流行？(Why is Deep Learning taking off?)
    • 第二周：神经网络的编程基础(Basics of Neural Network programming)
      • 2.1 二分类(Binary Classification)
      • 2.2 逻辑回归(Logistic Regression)
      • 2.3 逻辑回归的代价函数（Logistic Regression Cost Function）
      • 2.4 逻辑回归的梯度下降（Logistic Regression Gradient Descent）
      • 2.5 梯度下降的例子(Gradient Descent on m Examples)
      • 2.6 向量化 logistic 回归的梯度输出（Vectorizing Logistic Regression’s Gradient Output）
      • 2.7 （选修）logistic 损失函数的解释（Explanation of logistic regression cost function ）
      • Logistic Regression with a Neural Network mindset 代码
      • lr_utils.py
    • 第三周：浅层神经网络(Shallow neural networks)
      • 3.1 神经网络概述（Neural Network Overview）
      • 3.2 神经网络的表示（Neural Network Representation ）
      • 3.3 计算一个神经网络的输出（Computing a Neural Network's output ）
      • 3.4 多样本向量化（Vectorizing across multiple examples ）
      • 3.5 激活函数（Activation functions）
      • 3.6 为什么需要（ 非线性激活函数？（why need a nonlinear activation function?）
      • 3.7 激活函数的导数（Derivatives of activation functions ）
      • 3.8 神经网络的梯度下降（Gradient descent for neural networks）
      • 3.9 （选修）直观理解反向传播（Backpropagation intuition ）
      • 3.10 随机初始化（Random+Initialization）
      • Planar data classification with one hidden layer
      • planar_utils.py
      • testCases.py
    • 第四周：深层神经网络(Deep Neural Networks)
      • 4.1 深层神经网络（Deep L-layer neural network）
      • 4.2 前向传播和反向传播（Forward and backward propagation）
      • 4.3 深层网络中的前向传播（Forward propagation in a Deep Network ）
      • 4.4 为什么使用深层表示？（Why deep representations?）
      • 4.5 搭建神经网络块（Building blocks of deep neural networks）
      • 4.6 参数 VS 超参数（Parameters vs Hyperparameters）
      • Building your Deep Neural Network Step by Step
      • dnn_utils.py
      • testCases.py
      • Deep Neural Network Application
      • dnn_app_utils.py
  • 第二门课 改善深层神经网络：超参数调试、 正 则 化 以 及 优 化 (Improving Deep Neural Networks:Hyperparameter tuning, Regulariza
    • 第二门课 改善深层神经网络：超参数调试、正则化以及优化(Improving Deep Neural Networks:Hyperparameter tuning, Regularization and
      • 第一周：深度学习的实用层面(Practical aspects of Deep Learning)
        • 1.1 训练，验证，测试集（Train / Dev / Test sets）
        • 1.2 偏差，方差（Bias /Variance）
        • 1.3 机器学习基础（Basic Recipe for Machine Learning）
        • 1.4 正则化（Regularization）
        • 1.5 为什么正则化有利于预防过拟合呢？（Why regularization reduces overfitting?）
        • 1.6 dropout 正则化（Dropout Regularization）
        • 1.7 理解 dropout（Understanding Dropout）
        • 1.8 其他正则化方法（Other regularization methods）
        • 1.9 归一化输入（Normalizing inputs）
        • 1.10 梯度消失/梯度爆炸（Vanishing / Exploding gradients）
        • 1.11 神经网络的权重初始化（Weight Initialization for Deep Networks）
        • 1.12 梯度的数值逼近（Numerical approximation of gradients）
        • 1.13 梯度检验（Gradient checking）
        • 1.14 梯度检验应用的注意事项（Gradient Checking Implementation Notes）
        • Initialization
        • Gradient Checking
        • Regularization
        • reg_utils.py
        • testCases.py
      • 第二周：优化算法 (Optimization algorithms)
        • 2.1 Mini-batch 梯度下降（Mini-batch gradient descent）
        • 2.2 理解 mini-batch 梯度下降法（Understanding mini-batch gradient descent）
        • 2.3 指数加权平均数（Exponentially weighted averages）
        • 2.4 理解指数加权平均数（Understanding exponentially weighted averages ）
        • 2.5 指 数 加 权 平 均 的 偏 差 修 正 （ Bias correction in exponentially weighted averages ）
        • 2.6 动量梯度下降法（Gradient descent with Momentum ）
        • 2.7 RMSprop( root mean square prop)
        • 2.8 Adam 优化算法(Adam optimization algorithm)
        • 2.9 学习率衰减(Learning rate decay)
        • 2.10 局部最优的问题(The problem of local optima)
        • Optimization
        • opt_utils.py
        • testCases.py
      • 第 三 周 超 参 数 调 试 、 Batch 正 则 化 和 程 序 框 架 （Hyperparameter tuning）
        • 3.1 调试处理（Tuning process）
        • 3.2 为超参数选择合适的范围（Using an appropriate scale to pick hyperparameters）
        • 3.3 超参数训练的实践： Pandas VS Caviar（Hyperparameters tuning in practice: Pandas vs. Caviar）
        • 3.4 归一化网络的激活函数（ Normalizing activations in a network）
        • 3.5 将 Batch Norm 拟合进神经网络（Fitting Batch Norm into a neural network）
        • 3.6 Batch Norm 为什么奏效？（Why does Batch Norm work?）
        • 3.7 测试时的 Batch Norm（Batch Norm at test time）
        • 3.8 Softmax 回归（Softmax regression）
        • 3.9 训练一个 Softmax 分类器（Training a Softmax classifier）
        • tensorflow tutorial
        • improv_utils.py
        • tf_utils.py
  • 第三门课 结构化机器学习项目（Structuring Machine Learning Projects）
    • 第三门课 结构化机器学习项目（Structuring Machine Learning Projects）
      • 第一周 机器学习（ML）策略（1）（ML strategy（1））
        • 1.1 为什么是 ML 策略？（Why ML Strategy?）
        • 1.2 正交化（Orthogonalization）
        • 1.3 单一数字评估指标（Single number evaluation metric）
        • 1.4 满足和优化指标（Satisficing and optimizing metrics）
        • 1.5 训练/开发/测试集划分（Train/dev/test distributions）
        • 1.6 开发集和测试集的大小（Size of dev and test sets）
        • 1.7 什么时候该改变开发/测试集和指标？（When to change dev/test sets and metrics）
        • 1.8 为什么是人的表现？（ Why human-level performance?）
        • 1.9 可避免偏差（Avoidable bias）
        • 1.10 理解人的表现（Understanding human-level performance）
        • 1.11 超过人的表现（Surpassing human- level performance）
        • 1.12 改善你的模型的表现（Improving your model performance）
      • 第二周：机器学习策略（2）(ML Strategy (2))
        • 2.1 进行误差分析（Carrying out error analysis）
        • 2.2 清楚标注错误的数据（Cleaning up Incorrectly labeled data）
        • 2.3 快速搭建你的第一个系统，并进行迭代（Build your first system quickly, then iterate）
        • 2.4 在不同的划分上进行训练并测试（Training and testing on different distributions）
        • 2.5 不匹配数据划分的偏差和方差（Bias and Variance with mismatched data distributions）
        • 2.6 定位数据不匹配（Addressing data mismatch）
        • 2.7 迁移学习（Transfer learning）
        • 2.8 多任务学习（Multi-task learning）
        • 2.9 什么是端到端的深度学习？（What is end-to-end deep learning?）
        • 2.10 是否要使用端到端的深度学习？（Whether to use end-to-end learning?）
  • 第四门课 卷积神经网络（Convolutional Neural Networks）
    • 第四门课 卷积神经网络（Convolutional Neural Networks）
      • 第一周 卷积神经网络（Foundations of Convolutional Neural Networks）
        • 1.1 计算机视觉（Computer vision）
        • 1.2 边缘检测示例（Edge detection example）
        • 1.3 更多边缘检测内容（More edge detection）
        • 1.4 Padding
        • 1.5 卷积步长（Strided convolutions）
        • 1.6 三维卷积（Convolutions over volumes）
        • 1.7 单层卷积网络（One layer of a convolutional network）
        • 1.8 简单卷积网络示例（A simple convolution network example）
        • 1.9 池化层（Pooling layers）
        • 1.10 卷积神经网络示例（Convolutional neural network example）
        • 1.11 为什么使用卷积？（Why convolutions?）
        • Convolution model Step by Step
        • Convolutional Neural Networks: Application
        • cnn_utils
      • 第二周 深度卷积网络：实例探究（Deep convolutional models: case studies）
        • 2.1 经典网络（Classic networks）
        • 2.2 残差网络（Residual Networks (ResNets)）
        • 2.3 残差网络为什么有用？（Why ResNets work?）
        • 2.4 网络中的网络以及 1×1 卷积（Network in Network and 1×1 convolutions）
        • 2.5 谷歌 Inception 网络简介（Inception network motivation）
        • 2.6 Inception 网络（Inception network）
        • 2.7 迁移学习（Transfer Learning）
        • 2.8 数据扩充（Data augmentation）
        • 2.9 计算机视觉现状（The state of computer vision）
        • Residual Networks
        • Keras tutorial - the Happy House
        • kt_utils.py
      • 第三周 目标检测（Object detection）
        • 3.1 目标定位（Object localization）
        • 3.2 特征点检测（Landmark detection）
        • 3.3 目标检测（Object detection）
        • 3.4 卷积的滑动窗口实现（Convolutional implementation of sliding windows）
        • 3.5 Bounding Box预测（Bounding box predictions）
        • 3.6 交并比（Intersection over union）
        • 3.7 非极大值抑制（Non-max suppression）
        • 3.8 Anchor Boxes
        • 3.9 YOLO 算法（Putting it together: YOLO algorithm）
        • 3.10 候选区域（选修）（Region proposals (Optional)）
        • Autonomous driving application - Car detection
        • yolo_utils.py
      • 第四周 特殊应用：人脸识别和神经风格转换（Special applications: Face recognition &Neural style transfer）
        • 4.1 什么是人脸识别？（What is face recognition?）
        • 4.2 One-Shot学习（One-shot learning）
        • 4.3 Siamese 网络（Siamese network）
        • 4.4 Triplet 损失（Triplet 损失）
        • 4.5 面部验证与二分类（Face verification and binary classification）
        • 4.6 什么是深度卷积网络？（What are deep ConvNets learning?）
        • 4.7 代价函数（Cost function）
        • 4.8 内容代价函数（Content cost function）
        • 4.9 风格代价函数（Style cost function）
        • 4.10 一维到三维推广（1D and 3D generalizations of models）
        • Art Generation with Neural Style Transfer
        • nst_utils.py
        • Face Recognition for the Happy House
        • fr_utils.py
        • inception_blocks.py
  • 第五门课 序列模型(Sequence Models)
    • 第五门课 序列模型(Sequence Models)
      • 第一周 循环序列模型（Recurrent Neural Networks）
        • 1.1 为什么选择序列模型？（Why Sequence Models?）
        • 1.2 数学符号（Notation）
        • 1.3 循环神经网络模型（Recurrent Neural Network Model）
        • 1.4 通过时间的反向传播（Backpropagation through time）
        • 1.5 不同类型的循环神经网络（Different types of RNNs）
        • 1.6 语言模型和序列生成（Language model and sequence generation）
        • 1.7 对新序列采样（Sampling novel sequences）
        • 1.8 循环神经网络的梯度消失（Vanishing gradients with RNNs）
        • 1.9 GRU单元（Gated Recurrent Unit（GRU））
        • 1.10 长短期记忆（LSTM（long short term memory）unit）
        • 1.11 双向循环神经网络（Bidirectional RNN）
        • 1.12 深层循环神经网络（Deep RNNs）
        • Building your Recurrent Neural Network
        • rnn_utils.py
        • Dinosaurus Island -- Character level language model final
        • utils.py
        • shakespeare_utils.py
        • Improvise a Jazz Solo with an LSTM Network
      • 第二周 自然语言处理与词嵌入（Natural Language Processing and Word Embeddings）
        • 2.1 词汇表征（Word Representation）
        • 2.2 使用词嵌入（Using Word Embeddings）
        • 2.3 词嵌入的特性（Properties of Word Embeddings）
        • 2.4 嵌入矩阵（Embedding Matrix）
        • 2.5 学习词嵌入（Learning Word Embeddings）
        • 2.6 Word2Vec
        • 2.7 负采样（Negative Sampling）
        • 2.8 GloVe 词向量（GloVe Word Vectors）
        • 2.9 情感分类（Sentiment Classification）
        • 2.10 词嵌入除偏（Debiasing Word Embeddings）
        • Operations on word vectors
        • w2v_utils.py
        • Emojify
        • emo_utils.py
      • 第三周 序列模型和注意力机制（Sequence models & Attention mechanism）
        • 3.1 基础模型（Basic Models）
        • 3.2 选择最可能的句子（Picking the most likely sentence）
        • 3.3 集束搜索（Beam Search）
        • 3.4 改进集束搜索（Refinements to Beam Search）
        • 3.5 集束搜索的误差分析（Error analysis in beam search）
        • 3.6 Bleu 得分（选修）（Bleu Score (optional)）
        • 3.7 注意力模型直观理解（Attention Model Intuition）
        • 3.8注意力模型（Attention Model）
        • 3.9语音识别（Speech recognition）
        • 3.10触发字检测（Trigger Word Detection）
        • Neural machine translation with attention
        • nmt_utils.py
        • Trigger word detection
        • td_utils.py
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On this page
  • 1 - Packages
  • 2 - Outline of the Assignment
  • 3 - Convolutional Neural Networks
  • 3.1 - Zero-Padding
  • 3.2 - Single step of convolution
  • 3.3 - Convolutional Neural Networks - Forward pass
  • 4 - Pooling layer
  • 4.1 - Forward Pooling
  • 5 - Backpropagation in convolutional neural networks (OPTIONAL / UNGRADED)
  • 5.1 - Convolutional layer backward pass
  • 5.2 Pooling layer - backward pass
  • 5.2.1 Max pooling - backward pass
  • 5.2.2 - Average pooling - backward pass
  • 5.2.3 Putting it together: Pooling backward
  • Congratulations !

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  1. 第四门课 卷积神经网络（Convolutional Neural Networks）
  2. 第四门课 卷积神经网络（Convolutional Neural Networks）
  3. 第一周 卷积神经网络（Foundations of Convolutional Neural Networks）

Convolution model Step by Step

Previous1.11 为什么使用卷积？（Why convolutions?）NextConvolutional Neural Networks: Application

Last updated 6 years ago

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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][l] denotes an object of the lthl^{th}lth layer.

  • Example: a[4]a^{[4]}a[4] is the 4th4^{th}4th layer activation. W[5]W^{[5]}W[5] and b[5]b^{[5]}b[5] are the 5th5^{th}5th layer parameters.

  • Superscript (i)(i)(i) denotes an object from the ithi^{th}ith example.

  • Example: x(i)x^{(i)}x(i) is the ithi^{th}ith training example input.

  • Lowerscript iii denotes the ithi^{th}ith entry of a vector.

  • Example: ai[l]a^{[l]}_iai[l]​ denotes the ithi^{th}ith entry of the activations in layer lll, assuming this is a fully connected (FC) layer.

  • nHn_HnH​, nWn_WnW​ and nCn_CnC​ denote respectively the height, width and number of channels of a given layer. If you want to reference a specific layer lll, you can also write nH[l]n_H^{[l]}nH[l]​, nW[l]n_W^{[l]}nW[l]​, nC[l]n_C^{[l]}nC[l]​.

  • nHprevn_{H_{prev}}nHprev​​, nWprevn_{W_{prev}}nWprev​​ and nCprevn_{C_{prev}}nCprev​​ denote respectively the height, width and number of channels of the previous layer. If referencing a specific layer lll, this could also be denoted nH[l−1]n_H^{[l-1]}nH[l−1]​, nW[l−1]n_W^{[l-1]}nW[l−1]​, nC[l−1]n_C^{[l-1]}nC[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.

  • is the fundamental package for scientific computing with Python.

  • 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.

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)

The autoreload extension is already loaded. To reload it, use: %reload_ext autoreload

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:

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.

a = np.pad(a, ((0,0), (1,1), (0,0), (3,3), (0,0)), 'constant', constant_values = (..,..))
# 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])
print ("x_pad[1,1] =", x_pad[1,1])


fig, axarr = plt.subplots(1, 2)
axarr[0].set_title('x')
axarr[0].imshow(x[0,:,:,0])
axarr[1].set_title('x_pad')
axarr[1].imshow(x_pad[0,:,:,0])
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)

Figure 2 : 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.

# 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 = W*a_slice_prev
    # Sum over all entries of the volume s.
    Z = np.sum(s)
    # Add bias b to Z. Cast b to a float() so that Z results in a scalar value.
    Z = Z + float(b)
    ### END CODE HERE ###

    return Z
np.random.seed(1)
a_slice_prev = np.random.randn(4, 4, 3)
W = np.random.randn(4, 4, 3)
b = np.random.randn(1, 1, 1)


Z = conv_single_step(a_slice_prev, W, b)
print("Z =", Z)
Z = -6.999089450680221

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.

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.

# 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
    (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
np.random.seed(1)
A_prev = np.random.randn(10,4,4,3)
W = np.random.randn(2,2,3,8)
b = np.random.randn(1,1,1,8)
hparameters = {"pad" : 2,
"stride": 2}


Z, cache_conv = conv_forward(A_prev, W, b, hparameters)
print("Z's mean =", np.mean(Z))
print("Z[3,2,1] =", Z[3,2,1])
print("cache_conv[0][1][2][3] =", cache_conv[0][1][2][3])
Z's mean = 0.0489952035289
Z[3,2,1] = [-0.61490741 -6.7439236 -2.55153897 1.75698377 3.56208902 0.53036437
5.18531798 8.75898442]
cache_conv[0][1][2][3] = [-0.20075807 0.18656139 0.41005165]

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:

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:

# GRADED FUNCTION: pool_forward

def pool_forward(A_prev, hparameters, mode = "max"):
    """
    Implements the forward pass of the pooling layer

    Arguments:
    A_prev -- Input data, numpy array of shape (m, n_H_prev, n_W_prev, n_C_prev)
    hparameters -- python dictionary containing "f" and "stride"
    mode -- the pooling mode you would like to use, defined as a string ("max" or "average")

    Returns:
    A -- output of the pool layer, a numpy array of shape (m, n_H, n_W, n_C)
    cache -- cache used in the backward pass of the pooling layer, contains the input and hparameters 
    """

    # Retrieve dimensions from the input shape
    (m, n_H_prev, n_W_prev, n_C_prev) = A_prev.shape

    # Retrieve hyperparameters from "hparameters"
    f = hparameters["f"]
    stride = hparameters["stride"]

    # Define the dimensions of the output
    n_H = int(1 + (n_H_prev - f) / stride)
    n_W = int(1 + (n_W_prev - f) / stride)
    n_C = n_C_prev

    # Initialize output matrix A
    A = np.zeros((m, n_H, n_W, n_C))              

    ### START CODE HERE ###
    for i in range(m):                         # loop over the training examples
        for h in range(n_H):                     # loop on the vertical axis of the output volume
            for w in range(n_W):                 # loop on the horizontal axis of the output volume
                for c in range (n_C):            # loop over the channels of the output volume

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

                    # Use the corners to define the current slice on the ith training example of A_prev, channel c. (≈1 line)
                    a_prev_slice = A_prev[i,vert_start:vert_end,horiz_start:horiz_end,c]

                    # Compute the pooling operation on the slice. Use an if statment to differentiate the modes. Use np.max/np.mean.
                    if mode == "max":
                        A[i, h, w, c] = np.max(a_prev_slice)
                    elif mode == "average":
                        A[i,h,w,c]=np.mean(a_prev_slice)

    ### END CODE HERE ###

    # Store the input and hparameters in "cache" for pool_backward()
    cache = (A_prev, hparameters)

    # Making sure your output shape is correct
    assert(A.shape == (m, n_H, n_W, n_C))

    return A, cache
np.random.seed(1)
A_prev = np.random.randn(2, 4, 4, 3)
hparameters = {"stride" : 2, "f": 3}


A, cache = pool_forward(A_prev, hparameters)
print("mode = max")
print("A =", A)
print()
A, cache = pool_forward(A_prev, hparameters, mode = "average")
print("mode = average")
print("A =", A)
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]]]]

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:

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:

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:

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.

def conv_backward(dZ, cache):
    """
    Implement the backward propagation for a convolution function

    Arguments:
    dZ -- gradient of the cost with respect to the output of the conv layer (Z), numpy array of shape (m, n_H, n_W, n_C)
    cache -- cache of values needed for the conv_backward(), output of conv_forward()

    Returns:
    dA_prev -- gradient of the cost with respect to the input of the conv layer (A_prev),
               numpy array of shape (m, n_H_prev, n_W_prev, n_C_prev)
    dW -- gradient of the cost with respect to the weights of the conv layer (W)
          numpy array of shape (f, f, n_C_prev, n_C)
    db -- gradient of the cost with respect to the biases of the conv layer (b)
          numpy array of shape (1, 1, 1, n_C)
    """

    ### START CODE HERE ###
    # Retrieve information from "cache"
    (A_prev, W, b, hparameters) = cache

    # Retrieve dimensions from A_prev's shape
    (m, n_H_prev, n_W_prev, n_C_prev) = A_prev.shape

    # Retrieve dimensions from W's shape
    (f, f, n_C_prev, n_C) = W.shape

    # Retrieve information from "hparameters"
    stride = hparameters["stride"]
    pad = hparameters["pad"]

    # Retrieve dimensions from dZ's shape
    (m, n_H, n_W, n_C) = dZ.shape

    # Initialize dA_prev, dW, db with the correct shapes
    dA_prev = np.zeros((A_prev.shape))                           
    dW = np.zeros((W.shape))
    db = np.zeros((1,1,1,n_C))

    # Pad A_prev and dA_prev
    A_prev_pad = zero_pad(A_prev,pad)
    dA_prev_pad = zero_pad(dA_prev,pad)
    for i in range(m):                       # loop over the training examples

        # select ith training example from A_prev_pad and dA_prev_pad
        a_prev_pad = A_prev_pad[i,:,:,:]
        da_prev_pad = dA_prev_pad[i,:,:,:]

        for h in range(n_H):                   # loop over vertical axis of the output volume
            for w in range(n_W):               # loop over horizontal axis of the output volume
                for c in range(n_C):           # loop over the channels of the output volume

                    # Find the corners of the current "slice"
                    vert_start = h*stride
                    vert_end = h*stride+f
                    horiz_start = w*stride
                    horiz_end = w*stride+f

                    # Use the corners to define the slice from a_prev_pad
                    a_slice = a_prev_pad[vert_start:vert_end,horiz_start:horiz_end,:]

                    # Update gradients for the window and the filter's parameters using the code formulas given above
                    da_prev_pad[vert_start:vert_end, horiz_start:horiz_end, :] += W[:,:,:,c]*dZ[i,h,w,c]
                    dW[:,:,:,c] += a_slice*dZ[i,h,w,c]
                    db[:,:,:,c] += dZ[i,h,w,c]

        # Set the ith training example's dA_prev to the unpaded da_prev_pad (Hint: use X[pad:-pad, pad:-pad, :])
        dA_prev[i, :, :, :] = da_prev_pad[pad:-pad,pad:-pad,:]
    ### END CODE HERE ###

    # Making sure your output shape is correct
    assert(dA_prev.shape == (m, n_H_prev, n_W_prev, n_C_prev))

    return dA_prev, dW, db
np.random.seed(1)
dA, dW, db = conv_backward(Z, cache_conv)
print("dA_mean =", np.mean(dA))
print("dW_mean =", np.mean(dW))
print("db_mean =", np.mean(db))
dA_mean = 1.45243777754
dW_mean = 1.72699145831
db_mean = 7.83923256462

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:

  • 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.

def create_mask_from_window(x):
    """
    Creates a mask from an input matrix x, to identify the max entry of x.

    Arguments:
    x -- Array of shape (f, f)

    Returns:
    mask -- Array of the same shape as window, contains a True at the position corresponding to the max entry of x.
    """

    ### START CODE HERE ### (≈1 line)
    mask = (x==np.max(x))
    ### END CODE HERE ###

    return mask
np.random.seed(1)
x = np.random.randn(2,3)
mask = create_mask_from_window(x)
print('x = ', x)
print("mask = ", mask)
x = [[ 1.62434536 -0.61175641 -0.52817175]
     [-1.07296862 0.86540763 -2.3015387 ]]
mask = [[ True False False]
        [False False False]]

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:

def distribute_value(dz, shape):
    """
    Distributes the input value in the matrix of dimension shape

    Arguments:
    dz -- input scalar
    shape -- the shape (n_H, n_W) of the output matrix for which we want to distribute the value of dz

    Returns:
    a -- Array of size (n_H, n_W) for which we distributed the value of dz
    """

    ### START CODE HERE ###
    # Retrieve dimensions from shape (≈1 line)
    (n_H, n_W) = shape

    # Compute the value to distribute on the matrix (≈1 line)
    average = dz/(n_H*n_W)

    # Create a matrix where every entry is the "average" value (≈1 line)
    a = np.zeros((n_H,n_W))+average
    ### END CODE HERE ###

    return a
a = distribute_value(2, (2,2))
print('distributed value =', a)
distributed value = [[ 0.5 0.5]
                     [ 0.5 0.5]]

5.2.3 Putting it together: Pooling backward

You now have everything you need to compute backward propagation on a pooling layer.

Exercise: Implement the pool_backward function in both modes ("max" and "average"). You will once again use 4 for-loops (iterating over training examples, height, width, and channels). You should use an if/elif statement to see if the mode is equal to 'max' or 'average'. If it is equal to 'average' you should use the distribute_value() function you implemented above to create a matrix of the same shape as a_slice. Otherwise, the mode is equal to 'max', and you will create a mask with create_mask_from_window() and multiply it by the corresponding value of dZ.

def pool_backward(dA, cache, mode = "max"):
    """
    Implements the backward pass of the pooling layer

    Arguments:
    dA -- gradient of cost with respect to the output of the pooling layer, same shape as A
    cache -- cache output from the forward pass of the pooling layer, contains the layer's input and hparameters 
    mode -- the pooling mode you would like to use, defined as a string ("max" or "average")

    Returns:
    dA_prev -- gradient of cost with respect to the input of the pooling layer, same shape as A_prev
    """

    ### START CODE HERE ###

    # Retrieve information from cache (≈1 line)
    (A_prev, hparameters) = cache

    # Retrieve hyperparameters from "hparameters" (≈2 lines)
    stride = hparameters["stride"]
    f = hparameters["f"]

    # Retrieve dimensions from A_prev's shape and dA's shape (≈2 lines)
    m, n_H_prev, n_W_prev, n_C_prev = A_prev.shape
    m, n_H, n_W, n_C = dA.shape

    # Initialize dA_prev with zeros (≈1 line)
    dA_prev = np.zeros((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])
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.

Exercise: Implement the following function, which pads all the images of a batch of examples X with zeros. . Note if you want to pad the array "a" of shape (5,5,5,5,5)(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:

Exercise: Implement conv_single_step(). .

nH=⌊nHprev−f+2×padstride⌋+1n_H = \lfloor \frac{n_{H_{prev}} - f + 2 \times pad}{stride} \rfloor +1nH​=⌊stridenHprev​​−f+2×pad​⌋+1
nW=⌊nWprev−f+2×padstride⌋+1n_W = \lfloor \frac{n_{W_{prev}} - f + 2 \times pad}{stride} \rfloor +1nW​=⌊stridenWprev​​−f+2×pad​⌋+1

nC=number of filters used in the convolutionn_C = \text{number of filters used in the convolution}nC​=number of filters used in the convolution

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

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

nH=⌊nHprev−fstride⌋+1n_H = \lfloor \frac{n_{H_{prev}} - f}{stride} \rfloor +1nH​=⌊stridenHprev​​−f​⌋+1
nW=⌊nWprev−fstride⌋+1n_W = \lfloor \frac{n_{W_{prev}} - f}{stride} \rfloor +1nW​=⌊stridenWprev​​−f​⌋+1
nC=nCprevn_C = n_{C_{prev}}nC​=nCprev​​

This is the formula for computing dAdAdA with respect to the cost for a certain filter WcW_cWc​ and a given training example:

dA+=∑h=0nH∑w=0nWWc×dZhw                   (1)dA += \sum _{h=0} ^{n_H} \sum_{w=0} ^{n_W} W_c \times dZ_{hw} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)dA+=h=0∑nH​​w=0∑nW​​Wc​×dZhw​                   (1)

Where WcW_cWc​ is a filter and dZhwdZ_{hw}dZhw​ 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 WcW_cWc​ 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.

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

dWc+=∑h=0nH∑w=0nWaslice×dZhw                   (2)dW_c += \sum _{h=0} ^{n_H} \sum_{w=0} ^ {n_W} a_{slice} \times dZ_{hw} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)dWc​+=h=0∑nH​​w=0∑nW​​aslice​×dZhw​                   (2)

Where aslicea_{slice}aslice​ corresponds to the slice which was used to generate the acitivation ZijZ_{ij}Zij​. Hence, this ends up giving us the gradient for WWW with respect to that slice. Since it is the same WWW, we will just add up all such gradients to get dWdWdW.

This is the formula for computing dbdbdb with respect to the cost for a certain filter WcW_cWc​:

db=∑h∑wdZhw                   (3)db = \sum_h \sum_w dZ_{hw} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)db=h∑​w∑​dZhw​                   (3)

As you have previously seen in basic neural networks, db is computed by summing dZdZdZ. In this case, you are just summing over all the gradients of the conv output (Z) with respect to the cost.

X=[1342]→M=[0010]                   (4)X = \begin{bmatrix} 1 && 3 \\ 4 && 2 \end{bmatrix} \quad \rightarrow \quad M =\begin{bmatrix} 0 && 0 \\ 1 && 0 \end{bmatrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)X=[14​​32​]→M=[01​​00​]                   (4)

may be helpful. It computes the maximum of an array.

dZ=1→dZ=[1/41/41/41/4]                   (5)dZ = 1 \quad \rightarrow \quad dZ =\begin{bmatrix} 1/4 && 1/4 \\ 1/4 && 1/4 \end{bmatrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)dZ=1→dZ=[1/41/4​​1/41/4​]                   (5)

This implies that each position in the dZdZdZ 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
np.max()
Hint
numpy
matplotlib
Use np.pad