cs230 深度学习 Lecture 2 编程作业: Logistic Regression with a Neural Network mindset


1. 将 Logistic 表达为 神经网络 的形式

本文的目的是要用神经网络的思想实现 Logistic Regression,输入一张图片就可以判断该图片是不是猫。

那么什么是神经网络呢? 可以看我之前写的这篇文章:

什么是神经网络

其中一个很重要的概念,神经元:

再来看 Logistic 模型的表达:

那么把 Logistic 表达为 神经网络 的形式为:

(关于 Logistic 可以看这两篇文章: Logistic Regression 为什么用极大似然函数 Logistic regression 为什么用 sigmoid ?

接下来就可以构建模型:


2. 构建模型

我们的目的是学习 w 和 b 使 cost function J 达到最小,

方法就是:

  • 通过前向传播 (forward propagation) 计算当前的损失,
  • 通过反向传播 (backward propagation) 计算当前的梯度,
  • 再用梯度下降法对参数进行优化更新 (gradient descent)

关于反向传播可以看这两篇文章: 手写,纯享版反向传播算法公式推导

构建模型,训练模型,并进行预测,包含下面几步:

  1. 导入包
  2. 获得数据
  3. 并进行预处理: 格式转换,归一化
  4. 整合模型:
    • A. 构建模型
      • a. 初始化参数:w 和 b 为 0
      • b. 前向传播:计算当前的损失
      • c. 反向更新:计算当前的梯度
    • B. 梯度更新求模型参数
    • C. 进行预测
  5. 绘制学习曲线

下面进入详细代码:


1. 导入包

引入需要的 packages, 其中, h5py 是 python 中用于处理 H5 文件的接口, PIL 和 scipy 在本文是用自己的图片来测试训练好的模型, load_dataset 读取数据

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

其中 lr_utils.py 如下,是对 H5 文件进行解析 :

#lr_utils.py  

import numpy as np  
import h5py  
          
def load_dataset():  
    train_dataset = h5py.File('datasets/train_catvnoncat.h5', "r")  
    train_set_x_orig = np.array(train_dataset["train_set_x"][:]) # your train set features  
    train_set_y_orig = np.array(train_dataset["train_set_y"][:]) # your train set labels  
  
    test_dataset = h5py.File('datasets/test_catvnoncat.h5', "r")  
    test_set_x_orig = np.array(test_dataset["test_set_x"][:]) # your test set features  
    test_set_y_orig = np.array(test_dataset["test_set_y"][:]) # your test set labels  
  
    classes = np.array(test_dataset["list_classes"][:]) # the list of classes  
      
    train_set_y_orig = train_set_y_orig.reshape((1, train_set_y_orig.shape[0]))  
    test_set_y_orig = test_set_y_orig.reshape((1, test_set_y_orig.shape[0]))  
      
    return train_set_x_orig, train_set_y_orig, test_set_x_orig, test_set_y_orig, classes 

2. 获得数据

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

可以看一下图片的例子:

# 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.")

3. 进行预处理: 格式转换,归一化

这时需要获得下面几个值:

  • m_train (训练样本数量)
  • m_test (测试样本数量)
  • num_px (训练数据集的长和宽)
### START CODE HERE ### (≈ 3 lines of code)### STA 
m_train = train_set_y.shape[1]
m_test = test_set_y.shape[1]
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))

图像需要进行 reshape,原本是 (num_px, num_px, 3),要扁平化为一个向量 (num_px * num_px * 3, 1), 将 (a, b, c, d) 维的矩阵转换为 (b∗c∗d, a) 可以用: X_flatten = X.reshape(X.shape[0], -1).T

# 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]))

预处理还常常包括对数据进行中心化和标准化,图像数据的话,可以简单除以最大的像素值:

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

4. 整合模型

- A. 构建模型
    - a. 初始化参数:w 和 b 为 0
    - b. 前向传播:计算当前的损失
    - c. 反向更新:计算当前的梯度
- B. 梯度更新求模型参数
- C. 进行预测

先来 A. 构建模型

按照前面提到的三步: 初始化参数:w 和 b 为 0 前向传播:计算当前的损失 反向更新:计算当前的梯度

首先需要一个辅助函数 sigmoid( w^T x + b)

# GRADED FUNCTION: sigmoid

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

    Arguments:
    x -- 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

a. 初始化参数:w 和 b 为 0

# 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(shape=(dim, 1))
    b = 0
    ### END CODE HERE ###

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

b. 前向传播:计算当前的损失 c. 反向更新:计算当前的梯度

# 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
    """
    
    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 = (- 1 / m) * np.sum(Y * np.log(A) + (1 - Y) * (np.log(1 - A)))  # compute cost
    ### END CODE HERE ###
    
    # BACKWARD PROPAGATION (TO FIND GRAD)
    ### START CODE HERE ### (≈ 2 lines of code)
    dw = (1 / m) * np.dot(X, (A - Y).T)
    db = (1 / m) * np.sum(A - Y)
    ### 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

B. 梯度更新求模型参数

这一步 optimize 的目的是要学习 w 和 b 使 cost function J 达到最小, 用到的方法是梯度下降 \theta = \theta - \alpha \text{ } d\theta,

# 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  # need to broadcast
        b = b - learning_rate * db
        ### END CODE HERE ###
        
        # Record the costs
        if i % 100 == 0:
            costs.append(cost)
        
        # Print the cost every 100 training examples
        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

C. 进行预测

# 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)
        Y_prediction[0, i] = 1 if A[0, i] > 0.5 else 0
        ### END CODE HERE ###
    
    assert(Y_prediction.shape == (1, m))
    
    return Y_prediction

下面为整合的逻辑回归模型:

将参数初始化,优化求参,预测整合在一起,

输入为 训练集,测试集,迭代次数,学习速率,是否打印中间损失 打印 test 和 train 集的预测准确率 返回的 d 含有 参数 w,b,还有 test train 集上面的预测值,

# 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

下面代码进行模型训练:

d = model(train_set_x, train_set_y, test_set_x, test_set_y, num_iterations = 2000, learning_rate = 0.005, print_cost = True)

结果:

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 %

得到模型后可以看指定 index 所代表图片的预测值:

# Example of a picture that was wrongly classified.# Exampl 
index = 5
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.")

5. 绘制学习曲线

# Plot learning curve (with costs)# Plot l 
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()

可以看出 costs 是在下降的,如果增加迭代次数,那么训练数据的准确率会进一步提高,但是测试数据集的准确率可能会明显下降,这就是由于过拟合造成的。

还可以对比下不同学习率对应下的学习效果:

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')

legend = plt.legend(loc='upper center', shadow=True)
frame = legend.get_frame()
frame.set_facecolor('0.90')
plt.show()

当学习率过大 (例 0.01) 时,costs 出现上下震荡,甚至可能偏离,不过这里 0.01 最终幸运地收敛到了一个比较好的值。


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