深入浅出Logistic Regression之二分类


概述

在深度学习笔记-神经网络基础文章里面介绍过Logistic Regression模型进行二分类和推导。比如:我们现在有一张彩色图片让计算机自己识别图片中的动物是否是cat(猫)。下面我们深入讲解一下Logistic Regression的随机梯度反向传播算法的求解即db,dw,dz

单样本Logistic Regression的随机梯度下降算法(反向传播)

我们先来看一下单样本下的LR的随机梯度下降算法,如下图所示:

我们需要做的是根据随机梯度下降算法进行求解即反向传播算法来求解 下面是我自己计算并求随机梯度下降图片:

多样本Logistic Regression的随机梯度下降算法(反向传播)

我们将上述的数学计算表述为编程思想:

如果通过for循环来处理,那么对于我们的训练和工业应用来说会变得很不实际,那么我们下面通过向量化的方式来来解决这个问题。下面我们就通过编程实战来分析一下这个问题。

向量化的逻辑回归分类

实战逻辑回归二元分类

我们这边一个业务场景,需要分类自动识别一张图片的动物是否是猫(cat:y=1 or non-cat:y=0)。我们将图片的数据为了方便处理和使用,我们这边采用的是h5文件格式。这个数据集可以在网上找到,每个数据实例都是64643(width:64,height:64,RGB:3)图片,即(num_px, num_px, 3)。

1.加载数据集

import numpy as np
import matplotlib.pyplot as plt
import h5py
import scipy
from PIL import Image
from scipy import ndimage
def load_dataset():
    train_dataset = h5py.File('./train_catvnoncat.h5', "r")
	#训练数据集
    train_set_x_orig = np.array(train_dataset["train_set_x"][:])
	#训练数据集的类别
    train_set_y_orig = np.array(train_dataset["train_set_y"][:])
    test_dataset = h5py.File('./test_catvnoncat.h5', "r")
	#测试数据集
    test_set_x_orig = np.array(test_dataset["test_set_x"][:])
	#测试数据集的类别
    test_set_y_orig = np.array(test_dataset["test_set_y"][:])
	#所有数据的分类集合
    classes = np.array(test_dataset["list_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

下一步我们简单的看一下数据的结果,分析一下有多少训练数据和测试数据。

2.得到数据的信息和简单的数据处理

#训练数据集实例的size
m_train = train_set_x_orig.shape[0]
#测试数据集实例的size
m_test = test_set_x_orig.shape[0]
#每个实例的px
num_px = train_set_x_orig.shape[1]

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))
## 输出为
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)

X_flatten = X.reshape(X.shape[0], -1).T

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
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))
##输出
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)
#为了我们更方便的进行机器学习,我们需要对数据进行预处理,标准化数据。
train_set_x = train_set_x_flatten/255.
test_set_x = test_set_x_flatten/255.

3.开始建立我们算法模型

我们接下来开始定义模型和初始化模型参数,我们先定义一些简单的帮助函数。

3.1 帮助函数(sigmod function)和初始化参数模型

def sigmoid(z):
    """
    Compute the sigmoid of z
    Arguments:
    z -- A scalar or numpy array of any size.
    Return:
    s -- sigmoid(z)
    """
    s = 1/(1+np.exp(-z))
    return s
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)
    """
    
    w = np.zeros((dim,1))
    b = 0
	
    assert(w.shape == (dim, 1))
    assert(isinstance(b, float) or isinstance(b, int))
    
    return w, b

前向传播和反向传播梯度下降算法

根据我们之前的分析的梯度下降算法,我们依次来定义一下:

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)
    A = sigmoid(w.T.dot(X)+b)                                     # compute activation
    cost = -1/m*np.sum(np.dot(np.log(A),Y.T)+np.dot(np.log(1-A),(1-Y).T))                        # compute cost
    # BACKWARD PROPAGATION (TO FIND GRAD)
    dw = 1/m*np.dot(X,(A-Y).T)
    db = 1/m*np.sum(np.subtract(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
## 下面我们需要进行优化求解
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 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

进行预测

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
    A = sigmoid(np.dot(w.T,X)+b)
    Y_prediction=np.where(A<=0.5,0,1)
    assert(Y_prediction.shape == (1, m))
    
    return Y_prediction

最后我们需要定义一个model来进行封装。

model

# 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)
    dim = X_train.shape[0]
    w, b = initialize_with_zeros(dim)
    # Gradient descent (≈ 1 line of code)
    parameters, grads, costs = optimize(w, b, X_train, Y_train, num_iterations , learning_rate , print_cost )
    
     # Retrieve derivatives from grads
    dw = grads["dw"]
    db = grads["db"]
        
    # 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_train = predict(w,b,X_train)
    Y_prediction_test = predict(w,b,X_test)

    ### 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

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