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 - Overview of the Problem set
  • 3 - General Architecture of the learning algorithm
  • 4 - Building the parts of our algorithm
  • 4.1 - Helper functions
  • 4.2 - Initializing parameters
  • 4.3 - Forward and Backward propagation
  • 4.4 - Optimization
  • 5 - Merge all functions into a model
  • 6 - Further analysis (optional/ungraded exercise)
  • 7 - Test with your own image (optional/ungraded exercise)

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  1. 第一门课 神经网络和深度学习(Neural-Networks-and-Deep-Learning)
  2. 第二周:神经网络的编程基础(Basics of Neural Network programming)

Logistic Regression with a Neural Network mindset 代码

Welcome to your first (required) programming assignment! You will build a logistic regression classifier to recognize cats. This assignment will step you through how to do this with a Neural Network mindset, and so will also hone your intuitions about deep learning.

Instructions:

  • Do not use loops (for/while) in your code, unless the instructions explicitly ask you to do so.

You will learn to:

  • Build the general architecture of a learning algorithm, including:

    • Initializing parameters

    • Calculating the cost function and its gradient

    • Using an optimization algorithm (gradient descent)

  • Gather all three functions above into a main model function, in the right order.

1 - Packages

First, let's run the cell below to import all the packages that you will need during this assignment.

  • numpy is the fundamental package for scientific computing with Python.

  • h5py is a common package to interact with a dataset that is stored on an H5 file.

  • matplotlib is a famous library to plot graphs in Python.

  • PIL and scipy are used here to test your model with your own picture at the end.

import numpy as np
import matplotlib.pyplot as plt
import h5py
import scipy
from PIL import Image
from scipy import ndimage
from lr_utils import load_dataset

%matplotlib inline

2 - Overview of the Problem set

Problem Statement: You are given a dataset ("data.h5") containing:

  • a training set of m_train images labeled as cat (y=1) or non-cat (y=0)

  • a test set of m_test images labeled as cat or non-cat

  • each image is of shape (num_px, num_px, 3) where 3 is for the 3 channels (RGB). Thus, each image is square (height = num_px) and (width = num_px).

You will build a simple image-recognition algorithm that can correctly classify pictures as cat or non-cat.

Let's get more familiar with the dataset. Load the data by running the following code.

# Loading the data (cat/non-cat)
train_set_x_orig, train_set_y, test_set_x_orig, test_set_y, classes = load_dataset()

We added "_orig" at the end of image datasets (train and test) because we are going to preprocess them. After preprocessing, we will end up with train_set_x and test_set_x (the labels train_set_y and test_set_y don't need any preprocessing).

Each line of your train_set_x_orig and test_set_x_orig is an array representing an image. You can visualize an example by running the following code. Feel free also to change the index value and re-run to see other images.

# Example of a picture
index = 25
plt.imshow(train_set_x_orig[index])
print ("y = " + str(train_set_y[:, index]) + ", it's a '" + classes[np.squeeze(train_set_y[:, index])].decode("utf-8") +  "' picture.")
y = [1], it's a 'cat' picture.

Many software bugs in deep learning come from having matrix/vector dimensions that don't fit. If you can keep your matrix/vector dimensions straight you will go a long way toward eliminating many bugs.

Exercise: Find the values for:

  • m_train (number of training examples)

  • m_test (number of test examples)

  • num_px (= height = width of a training image)

    Remember that train_set_x_orig is a numpy-array of shape (m_train, num_px, num_px, 3). For instance, you can access m_train by writing train_set_x_orig.shape[0].

### START CODE HERE ### (≈ 3 lines of code)
m_train = train_set_x_orig.shape[0]
m_test = test_set_x_orig.shape[0]
num_px = train_set_x_orig.shape[1]
### END CODE HERE ###

print ("Number of training examples: m_train = " + str(m_train))
print ("Number of testing examples: m_test = " + str(m_test))
print ("Height/Width of each image: num_px = " + str(num_px))
print ("Each image is of size: (" + str(num_px) + ", " + str(num_px) + ", 3)")
print ("train_set_x shape: " + str(train_set_x_orig.shape))
print ("train_set_y shape: " + str(train_set_y.shape))
print ("test_set_x shape: " + str(test_set_x_orig.shape))
print ("test_set_y shape: " + str(test_set_y.shape))
Number of training examples: m_train = 209
Number of testing examples: m_test = 50
Height/Width of each image: num_px = 64
Each image is of size: (64, 64, 3)
train_set_x shape: (209, 64, 64, 3)
train_set_y shape: (1, 209)
test_set_x shape: (50, 64, 64, 3)
test_set_y shape: (1, 50)

For convenience, you should now reshape images of shape (num_px, num_px, 3) in a numpy-array of shape (num_px*num_px*3, 1). After this, our training (and test) dataset is a numpy-array where each column represents a flattened image. There should be m_train (respectively m_test) columns.

Exercise: Reshape the training and test data sets so that images of size (num_px, num_px, 3) are flattened into single vectors of shape (num_px ∗*∗ num_px ∗*∗ 3, 1).

A trick when you want to flatten a matrix X of shape (a,b,c,d) to a matrix X_flatten of shape (b∗*∗c∗*∗d, a) is to use:

X_flatten = X.reshape(X.shape[0], -1).T      # X.T is the transpose of X
# Reshape the training and test examples

### START CODE HERE ### (≈ 2 lines of code)
train_set_x_flatten = train_set_x_orig.reshape(train_set_x_orig.shape[0],-1).T
test_set_x_flatten = test_set_x_orig.reshape(test_set_x_orig.shape[0],-1).T
### END CODE HERE ###

print ("train_set_x_flatten shape: " + str(train_set_x_flatten.shape))
print ("train_set_y shape: " + str(train_set_y.shape))
print ("test_set_x_flatten shape: " + str(test_set_x_flatten.shape))
print ("test_set_y shape: " + str(test_set_y.shape))
print ("sanity check after reshaping: " + str(train_set_x_flatten[0:5,0]))
train_set_x_flatten shape: (12288, 209)
train_set_y shape: (1, 209)
test_set_x_flatten shape: (12288, 50)
test_set_y shape: (1, 50)
sanity check after reshaping: [17 31 56 22 33]

To represent color images, the red, green and blue channels (RGB) must be specified for each pixel, and so the pixel value is actually a vector of three numbers ranging from 0 to 255.

One common preprocessing step in machine learning is to center and standardize your dataset, meaning that you substract the mean of the whole numpy array from each example, and then divide each example by the standard deviation of the whole numpy array. But for picture datasets, it is simpler and more convenient and works almost as well to just divide every row of the dataset by 255 (the maximum value of a pixel channel).

Let's standardize our dataset.

train_set_x = train_set_x_flatten/255.
test_set_x = test_set_x_flatten/255.

What you need to remember:

Common steps for pre-processing a new dataset are:

  • Figure out the dimensions and shapes of the problem (m_train, m_test, num_px, ...)

  • Reshape the datasets such that each example is now a vector of size (num_px * num_px * 3, 1)

  • "Standardize" the data

3 - General Architecture of the learning algorithm

It's time to design a simple algorithm to distinguish cat images from non-cat images.

You will build a Logistic Regression, using a Neural Network mindset. The following Figure explains why Logistic Regression is actually a very simple Neural Network!

For one example x(i)x^{(i)}x(i):

z(i)=wTx(i)+b      (1)z^{(i)} = w^T x^{(i)} + b \ \ \ \ \ \ (1)z(i)=wTx(i)+b      (1)
y^(i)=a(i)=sigmoid(z(i))      (2)\hat{y}^{(i)} = a^{(i)} = sigmoid(z^{(i)})\ \ \ \ \ \ (2)y^​(i)=a(i)=sigmoid(z(i))      (2)
L(a(i),y(i))=−y(i)log⁡(a(i))−(1−y(i))log⁡(1−a(i))      (3)\mathcal{L}(a^{(i)}, y^{(i)}) = - y^{(i)} \log(a^{(i)}) - (1-y^{(i)} ) \log(1-a^{(i)})\ \ \ \ \ \ (3)L(a(i),y(i))=−y(i)log(a(i))−(1−y(i))log(1−a(i))      (3)

The cost is then computed by summing over all training examples:

J=1m∑i=1mL(a(i),y(i))      (4)J = \frac{1}{m} \sum_{i=1}^m \mathcal{L}(a^{(i)}, y^{(i)})\ \ \ \ \ \ (4)J=m1​i=1∑m​L(a(i),y(i))      (4)

Key steps: In this exercise, you will carry out the following steps:

  • Initialize the parameters of the model

  • Learn the parameters for the model by minimizing the cost

  • Use the learned parameters to make predictions (on the test set)

  • Analyse the results and conclude

4 - Building the parts of our algorithm

The main steps for building a Neural Network are: 1. Define the model structure (such as number of input features) 2. Initialize the model's parameters 3. Loop:

  • Calculate current loss (forward propagation)

  • Calculate current gradient (backward propagation)

  • Update parameters (gradient descent)

You often build 1-3 separately and integrate them into one function we call model().

4.1 - Helper functions

Exercise: Using your code from "Python Basics", implement sigmoid(). As you've seen in the figure above, you need to compute sigmoid(wTx+b)=11+e−(wTx+b)sigmoid( w^T x + b) = \frac{1}{1 + e^{-(w^T x + b)}}sigmoid(wTx+b)=1+e−(wTx+b)1​ to make predictions. Use np.exp().

# GRADED FUNCTION: sigmoid

def sigmoid(z):
    """
    Compute the sigmoid of z

    Arguments:
    z -- A scalar or numpy array of any size.

    Return:
    s -- sigmoid(z)
    """

    ### START CODE HERE ### (≈ 1 line of code)
    s = 1/(1 + np.exp(-z))
    ### END CODE HERE ###

    return s
print ("sigmoid([0, 2]) = " + str(sigmoid(np.array([0,2]))))
sigmoid([0, 2]) = [ 0.5         0.88079708]

4.2 - Initializing parameters

Exercise: Implement parameter initialization in the cell below. You have to initialize w as a vector of zeros. If you don't know what numpy function to use, look up np.zeros() in the Numpy library's documentation.

# GRADED FUNCTION: initialize_with_zeros

def initialize_with_zeros(dim):
    """
    This function creates a vector of zeros of shape (dim, 1) for w and initializes b to 0.

    Argument:
    dim -- size of the w vector we want (or number of parameters in this case)

    Returns:
    w -- initialized vector of shape (dim, 1)
    b -- initialized scalar (corresponds to the bias)
    """

    ### START CODE HERE ### (≈ 1 line of code)
    w = np.zeros((dim,1))
    b = 0
    ### END CODE HERE ###

    assert(w.shape == (dim, 1))
    assert(isinstance(b, float) or isinstance(b, int))

    return w, b
dim = 2
w, b = initialize_with_zeros(dim)
print ("w = " + str(w))
print ("b = " + str(b))
w = [[ 0.]
     [ 0.]]

b = 0

For image inputs, w will be of shape (num_px ×\times× num_px ×\times× 3, 1).

4.3 - Forward and Backward propagation

Now that your parameters are initialized, you can do the "forward" and "backward" propagation steps for learning the parameters.

Exercise: Implement a function propagate() that computes the cost function and its gradient.

Hints:

Forward Propagation:

  • You get X

  • You compute A=σ(wTX+b)=(a(1),a(2),...,a(m−1),a(m))A = \sigma(w^T X + b) = (a^{(1)}, a^{(2)}, ..., a^{(m-1)}, a^{(m)})A=σ(wTX+b)=(a(1),a(2),...,a(m−1),a(m))

  • You calculate the cost function: J=−1m∑i=1my(i)log⁡(a(i))+(1−y(i))log⁡(1−a(i))J = -\frac{1}{m}\sum_{i=1}^{m}y^{(i)}\log(a^{(i)})+(1-y^{(i)})\log(1-a^{(i)})J=−m1​∑i=1m​y(i)log(a(i))+(1−y(i))log(1−a(i))

Here are the two formulas you will be using:

∂J∂w=1mX(A−Y)T      (5)\frac{\partial J}{\partial w} = \frac{1}{m}X(A-Y)^T\ \ \ \ \ \ (5)∂w∂J​=m1​X(A−Y)T      (5)
∂J∂b=1m∑i=1m(a(i)−y(i))      (6)\frac{\partial J}{\partial b} = \frac{1}{m} \sum_{i=1}^m (a^{(i)}-y^{(i)})\ \ \ \ \ \ (6)∂b∂J​=m1​i=1∑m​(a(i)−y(i))      (6)
# GRADED FUNCTION: propagate

def propagate(w, b, X, Y):
    """
    Implement the cost function and its gradient for the propagation explained above

    Arguments:
    w -- weights, a numpy array of size (num_px * num_px * 3, 1)
    b -- bias, a scalar
    X -- data of size (num_px * num_px * 3, number of examples)
    Y -- true "label" vector (containing 0 if non-cat, 1 if cat) of size (1, number of examples)

    Return:
    cost -- negative log-likelihood cost for logistic regression
    dw -- gradient of the loss with respect to w, thus same shape as w
    db -- gradient of the loss with respect to b, thus same shape as b

    Tips:
    - Write your code step by step for the propagation. np.log(), np.dot()
    """

    m = X.shape[1]

    # FORWARD PROPAGATION (FROM X TO COST)
    ### START CODE HERE ### (≈ 2 lines of code)
    A = sigmoid(np.dot(w.T, X) + b)                                # compute activation
    cost = -(Y*np.log(A) + (1 - Y)*np.log(1 - A)).sum()/m                                # compute cost
    ### END CODE HERE ###

    # BACKWARD PROPAGATION (TO FIND GRAD)
    ### START CODE HERE ### (≈ 2 lines of code)
    dw = np.dot(X, (A - Y).T)/m
    db = (A-Y).sum()/m
    ### END CODE HERE ###

    assert(dw.shape == w.shape)
    assert(db.dtype == float)
    cost = np.squeeze(cost)
    assert(cost.shape == ())

    grads = {"dw": dw,
             "db": db}

    return grads, cost
w, b, X, Y = np.array([[1.],[2.]]), 2., np.array([[1.,2.,-1.],[3.,4.,-3.2]]), np.array([[1,0,1]])
grads, cost = propagate(w, b, X, Y)
print ("dw = " + str(grads["dw"]))
print ("db = " + str(grads["db"]))
print ("cost = " + str(cost))
dw = [[ 0.99845601]
      [ 2.39507239]]

db = 0.00145557813678

cost = 5.80154531939

4.4 - Optimization

  • You have initialized your parameters.

  • You are also able to compute a cost function and its gradient.

  • Now, you want to update the parameters using gradient descent.

Exercise: Write down the optimization function. The goal is to learn www and bbb by minimizing the cost function JJJ. For a parameter θ\thetaθ, the update rule is θ=θ−α dθ\theta = \theta - \alpha \text{ } d\thetaθ=θ−α dθ, where α\alphaα is the learning rate.

# GRADED FUNCTION: optimize

def optimize(w, b, X, Y, num_iterations, learning_rate, print_cost = False):
    """
    This function optimizes w and b by running a gradient descent algorithm

    Arguments:
    w -- weights, a numpy array of size (num_px * num_px * 3, 1)
    b -- bias, a scalar
    X -- data of shape (num_px * num_px * 3, number of examples)
    Y -- true "label" vector (containing 0 if non-cat, 1 if cat), of shape (1, number of examples)
    num_iterations -- number of iterations of the optimization loop
    learning_rate -- learning rate of the gradient descent update rule
    print_cost -- True to print the loss every 100 steps

    Returns:
    params -- dictionary containing the weights w and bias b
    grads -- dictionary containing the gradients of the weights and bias with respect to the cost function
    costs -- list of all the costs computed during the optimization, this will be used to plot the learning curve.

    Tips:
    You basically need to write down two steps and iterate through them:
        1) Calculate the cost and the gradient for the current parameters. Use propagate().
        2) Update the parameters using gradient descent rule for w and b.
    """

    costs = []

    for i in range(num_iterations):


        # Cost and gradient calculation (≈ 1-4 lines of code)
        ### START CODE HERE ### 
        grads, cost = propagate(w, b, X, Y)
        ### END CODE HERE ###

        # Retrieve derivatives from grads
        dw = grads["dw"]
        db = grads["db"]

        # update rule (≈ 2 lines of code)
        ### START CODE HERE ###
        w = w - learning_rate*dw
        b = b - learning_rate*db
        ### END CODE HERE ###

        # Record the costs
        if i % 100 == 0:
            costs.append(cost)

        # Print the cost every 100 training iterations
        if print_cost and i % 100 == 0:
            print ("Cost after iteration %i: %f" %(i, cost))

    params = {"w": w,
              "b": b}

    grads = {"dw": dw,
             "db": db}

    return params, grads, costs
params, grads, costs = optimize(w, b, X, Y, num_iterations= 100, learning_rate = 0.009, print_cost = False)

print ("w = " + str(params["w"]))
print ("b = " + str(params["b"]))
print ("dw = " + str(grads["dw"]))
print ("db = " + str(grads["db"]))
w = [[ 0.19033591]
     [ 0.12259159]]

b = 1.92535983008

dw = [[ 0.67752042]
      [ 1.41625495]]

db = 0.219194504541

Exercise: The previous function will output the learned w and b. We are able to use w and b to predict the labels for a dataset X. Implement the predict() function. There are two steps to computing predictions:

  1. Calculate Y^=A=σ(wTX+b)\hat{Y} = A = \sigma(w^T X + b)Y^=A=σ(wTX+b)

  2. Convert the entries of a into 0 (if activation <= 0.5) or 1 (if activation > 0.5), stores the predictions in a vector Y_prediction. If you wish, you can use an if/else statement in a for loop (though there is also a way to vectorize this).

# GRADED FUNCTION: predict

def predict(w, b, X):
    '''
    Predict whether the label is 0 or 1 using learned logistic regression parameters (w, b)

    Arguments:
    w -- weights, a numpy array of size (num_px * num_px * 3, 1)
    b -- bias, a scalar
    X -- data of size (num_px * num_px * 3, number of examples)

    Returns:
    Y_prediction -- a numpy array (vector) containing all predictions (0/1) for the examples in X
    '''

    m = X.shape[1]
    Y_prediction = np.zeros((1,m))
    w = w.reshape(X.shape[0], 1)

    # Compute vector "A" predicting the probabilities of a cat being present in the picture
    ### START CODE HERE ### (≈ 1 line of code)
    A = sigmoid(np.dot(w.T, X) + b)
    ### END CODE HERE ###

    for i in range(A.shape[1]):

        # Convert probabilities A[0,i] to actual predictions p[0,i]
        ### START CODE HERE ### (≈ 4 lines of code)
        if A[:,i]>.5:
            Y_prediction[:,i] = 1
        ### END CODE HERE ###

    assert(Y_prediction.shape == (1, m))

    return Y_prediction
w = np.array([[0.1124579],[0.23106775]])
b = -0.3
X = np.array([[1.,-1.1,-3.2],[1.2,2.,0.1]])
print ("predictions = " + str(predict(w, b, X)))
predictions = [[ 1.  1.  0.]]

What to remember: You've implemented several functions that:

  • Initialize (w,b)

  • Optimize the loss iteratively to learn parameters (w,b):

    • computing the cost and its gradient

    • updating the parameters using gradient descent

  • Use the learned (w,b) to predict the labels for a given set of examples

5 - Merge all functions into a model

You will now see how the overall model is structured by putting together all the building blocks (functions implemented in the previous parts) together, in the right order.

Exercise: Implement the model function. Use the following notation:

  • Y_prediction_test for your predictions on the test set

  • Y_prediction_train for your predictions on the train set

  • w, costs, grads for the outputs of optimize()

# GRADED FUNCTION: model

def model(X_train, Y_train, X_test, Y_test, num_iterations = 2000, learning_rate = 0.5, print_cost = False):
    """
    Builds the logistic regression model by calling the function you've implemented previously

    Arguments:
    X_train -- training set represented by a numpy array of shape (num_px * num_px * 3, m_train)
    Y_train -- training labels represented by a numpy array (vector) of shape (1, m_train)
    X_test -- test set represented by a numpy array of shape (num_px * num_px * 3, m_test)
    Y_test -- test labels represented by a numpy array (vector) of shape (1, m_test)
    num_iterations -- hyperparameter representing the number of iterations to optimize the parameters
    learning_rate -- hyperparameter representing the learning rate used in the update rule of optimize()
    print_cost -- Set to true to print the cost every 100 iterations

    Returns:
    d -- dictionary containing information about the model.
    """

    ### START CODE HERE ###

    # initialize parameters with zeros (≈ 1 line of code)
    w, b = initialize_with_zeros(X_train.shape[0])

    # Gradient descent (≈ 1 line of code)
    parameters, grads, costs = optimize(w, b, X_train, Y_train, num_iterations, learning_rate, print_cost)

    # Retrieve parameters w and b from dictionary "parameters"
    w = parameters["w"]
    b = parameters["b"]

    # Predict test/train set examples (≈ 2 lines of code)
    Y_prediction_test = predict(w, b, X_test)
    Y_prediction_train = predict(w ,b, X_train)

    ### END CODE HERE ###

    # Print train/test Errors
    print("train accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_train - Y_train)) * 100))
    print("test accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_test - Y_test)) * 100))


    d = {"costs": costs,
         "Y_prediction_test": Y_prediction_test, 
         "Y_prediction_train" : Y_prediction_train, 
         "w" : w, 
         "b" : b,
         "learning_rate" : learning_rate,
         "num_iterations": num_iterations}

    return d

Run the following cell to train your model.

d = model(train_set_x, train_set_y, test_set_x, test_set_y, num_iterations = 2000, learning_rate = 0.005, print_cost = True)
Cost after iteration 0: 0.693147
Cost after iteration 100: 0.584508
Cost after iteration 200: 0.466949
Cost after iteration 300: 0.376007
Cost after iteration 400: 0.331463
Cost after iteration 500: 0.303273
Cost after iteration 600: 0.279880
Cost after iteration 700: 0.260042
Cost after iteration 800: 0.242941
Cost after iteration 900: 0.228004
Cost after iteration 1000: 0.214820
Cost after iteration 1100: 0.203078
Cost after iteration 1200: 0.192544
Cost after iteration 1300: 0.183033
Cost after iteration 1400: 0.174399
Cost after iteration 1500: 0.166521
Cost after iteration 1600: 0.159305
Cost after iteration 1700: 0.152667
Cost after iteration 1800: 0.146542
Cost after iteration 1900: 0.140872
train accuracy: 99.04306220095694 %
test accuracy: 70.0 %

Comment: Training accuracy is close to 100%. This is a good sanity check: your model is working and has high enough capacity to fit the training data. Test error is 68%. It is actually not bad for this simple model, given the small dataset we used and that logistic regression is a linear classifier. But no worries, you'll build an even better classifier next week!

Also, you see that the model is clearly overfitting the training data. Later in this specialization you will learn how to reduce overfitting, for example by using regularization. Using the code below (and changing the index variable) you can look at predictions on pictures of the test set.

# Example of a picture that was wrongly classified.
index = 1
plt.imshow(test_set_x[:,index].reshape((num_px, num_px, 3)))
print ("y = " + str(test_set_y[0,index]) + ", you predicted that it is a \"" + classes[d["Y_prediction_test"][0,index]].decode("utf-8") +  "\" picture.")
y = 1, you predicted that it is a "cat" picture.

Let's also plot the cost function and the gradients.

# Plot learning curve (with costs)
costs = np.squeeze(d['costs'])
plt.plot(costs)
plt.ylabel('cost')
plt.xlabel('iterations (per hundreds)')
plt.title("Learning rate =" + str(d["learning_rate"]))
plt.show()

Interpretation: You can see the cost decreasing. It shows that the parameters are being learned. However, you see that you could train the model even more on the training set. Try to increase the number of iterations in the cell above and rerun the cells. You might see that the training set accuracy goes up, but the test set accuracy goes down. This is called overfitting.

6 - Further analysis (optional/ungraded exercise)

Congratulations on building your first image classification model. Let's analyze it further, and examine possible choices for the learning rate α\alphaα.

Choice of learning rate

Reminder: In order for Gradient Descent to work you must choose the learning rate wisely. The learning rate α\alphaα determines how rapidly we update the parameters. If the learning rate is too large we may "overshoot" the optimal value. Similarly, if it is too small we will need too many iterations to converge to the best values. That's why it is crucial to use a well-tuned learning rate.

Let's compare the learning curve of our model with several choices of learning rates. Run the cell below. This should take about 1 minute. Feel free also to try different values than the three we have initialized the learning_rates variable to contain, and see what happens.

learning_rates = [0.01, 0.001, 0.0001]
models = {}
for i in learning_rates:
    print ("learning rate is: " + str(i))
    models[str(i)] = model(train_set_x, train_set_y, test_set_x, test_set_y, num_iterations = 1500, learning_rate = i, print_cost = False)
    print ('\n' + "-------------------------------------------------------" + '\n')

for i in learning_rates:
    plt.plot(np.squeeze(models[str(i)]["costs"]), label= str(models[str(i)]["learning_rate"]))

plt.ylabel('cost')
plt.xlabel('iterations (hundreds)')

legend = plt.legend(loc='upper center', shadow=True)
frame = legend.get_frame()
frame.set_facecolor('0.90')
plt.show()
learning rate is: 0.01
train accuracy: 99.52153110047847 %
test accuracy: 68.0 %

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

learning rate is: 0.001
train accuracy: 88.99521531100478 %
test accuracy: 64.0 %

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

learning rate is: 0.0001
train accuracy: 68.42105263157895 %
test accuracy: 36.0 %

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

Interpretation:

  • Different learning rates give different costs and thus different predictions results.

  • If the learning rate is too large (0.01), the cost may oscillate up and down. It may even diverge (though in this example, using 0.01 still eventually ends up at a good value for the cost).

  • A lower cost doesn't mean a better model. You have to check if there is possibly overfitting. It happens when the training accuracy is a lot higher than the test accuracy.

  • In deep learning, we usually recommend that you:

    • Choose the learning rate that better minimizes the cost function.

    • If your model overfits, use other techniques to reduce overfitting. (We'll talk about this in later videos.)

7 - Test with your own image (optional/ungraded exercise)

Congratulations on finishing this assignment. You can use your own image and see the output of your model. To do that: 1. Click on "File" in the upper bar of this notebook, then click "Open" to go on your Coursera Hub. 2. Add your image to this Jupyter Notebook's directory, in the "images" folder 3. Change your image's name in the following code 4. Run the code and check if the algorithm is right (1 = cat, 0 = non-cat)!

## START CODE HERE ## (PUT YOUR IMAGE NAME) 
my_image = "my_image.jpg"   # change this to the name of your image file 
## END CODE HERE ##

# We preprocess the image to fit your algorithm.
fname = "images/" + my_image
image = np.array(ndimage.imread(fname, flatten=False))
my_image = scipy.misc.imresize(image, size=(num_px,num_px)).reshape((1, num_px*num_px*3)).T
my_predicted_image = predict(d["w"], d["b"], my_image)

plt.imshow(image)
print("y = " + str(np.squeeze(my_predicted_image)) + ", your algorithm predicts a \"" + classes[int(np.squeeze(my_predicted_image)),].decode("utf-8") +  "\" picture.")
y = 0.0, your algorithm predicts a "non-cat" picture.

What to remember from this assignment: 1. Preprocessing the dataset is important. 2. You implemented each function separately: initialize(), propagate(), optimize(). Then you built a model(). 3. Tuning the learning rate (which is an example of a "hyperparameter") can make a big difference to the algorithm. You will see more examples of this later in this course!

Finally, if you'd like, we invite you to try different things on this Notebook. Make sure you submit before trying anything. Once you submit, things you can play with include:

  • Play with the learning rate and the number of iterations

  • Try different initialization methods and compare the results

  • Test other preprocessings (center the data, or divide each row by its standard deviation)

Bibliography:

  • http://www.wildml.com/2015/09/implementing-a-neural-network-from-scratch/

  • https://stats.stackexchange.com/questions/211436/why-do-we-normalize-images-by-subtracting-the-datasets-image-mean-and-not-the-c

Previous2.7 (选修)logistic 损失函数的解释(Explanation of logistic regression cost function )Nextlr_utils.py

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Mathematical expression of the algorithm:

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