Logistic Regression with a Neural Network mindset

Posted by LudoArt on February 8, 2019

1 导入所需的包

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

包介绍

  • 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.
  • lr_utils是压缩包内给的自定义函数,功能是导入数据集

PS:遇到的几个坑 坑1:PIL包不支持python3以上 解决方案:安装PIL的分支Pillow

安装完成后,使用from PIL import Image就引用使用库了。比如:

from PIL import Image
im = Image.open("bride.jpg")
im.rotate(45).show()

参考链接:https://blog.csdn.net/dcz1994/article/details/71642979

坑2:%matplotlib inline在PyCharm中报错 解决方案:使用from matplotlib import pyplot as plt

故导入包的代码可改为以下代码

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

参考链接:https://blog.csdn.net/xinluqishi123/article/details/63523531

2 数据预处理

数据预处理的一般步骤

  • 弄清楚要处理的数据集的大小和形状
  • 重塑一些数据集的形状
  • “标准化”数据

具体步骤如下

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

2.1 识别数据集的大小和形状

# Figure out the dimensions and shapes of the problem (m_train, m_test, num_px, ...)
m_train = train_set_x_orig.shape[0]
m_test = test_set_x_orig.shape[0]
num_px = train_set_x_orig.shape[1]

2.2 重塑一些数据的形状

# Reshape the training and test examples
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

2.3 “标准化”数据

# standardize our dataset.
train_set_x = train_set_x_flatten/255.
test_set_x = test_set_x_flatten/255.

3 学习算法的一般体系结构

Logistic Regression算法示意图如下: Logistic Regression算法示意图 Logistic Regression算法的数学表达式 Logistic Regression算法的数学表达式 Logistic Regression算法的关键步骤

  • 初始化模型的参数
  • 通过最小化损失来学习模型的参数
  • 使用学习的参数在测试集上进行预测
  • 分析结果并得出结论

4 构建Logistic Regression算法的各个部分

构建一个神经网络的主要步骤如下: 1、定义模型的结构 2、初始化模型的参数 3、循环

  • 计算当前的损失(正向传播过程)
  • 计算当前的梯度(反向传播过程)
  • 更新参数(梯度下降)

通常单独构建1-3并将它们集成到一个我们称之为model()的函数中。

4.1 sigmoid函数

# 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)
    """

    s = 1/(1+np.exp(-z))
    return s

4.2 初始化参数

将参数w和b都初始化为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)
    """
    w = np.zeros((dim, 1))
    b = 0
    assert (w.shape == (dim, 1))
    assert (isinstance(b, float) or isinstance(b, int))
    return w, b

4.3 前向传播和反向传播过程

正向传播过程:

  • 获取数据集X
  • 计算
  • 计算成本函数:

反向传播过程:

# 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)
    A = sigmoid(np.dot(w.T, X)+b)  # compute activation
    cost = -np.sum((np.dot(Y, np.log(A).T)+np.dot(1-Y, np.log(1-A).T)))/m  # compute cost

    # BACKWARD PROPAGATION (TO FIND GRAD)
    dw = (np.dot(X, (A-Y).T))/m
    db = (np.sum(A-Y))/m

    assert (dw.shape == w.shape)
    assert (db.dtype == float)
    cost = np.squeeze(cost)
    assert (cost.shape == ())
    grads = {"dw": dw, "db": db}
    return grads, cost

4.4 优化

  • 已对参数做了初始化操作
  • 已可进行成本函数和梯度的计算
  • 现在,使用梯度下降来更新参数
# 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
        grads, cost = propagate(w, b, X, Y)

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

        # update rule
        w = w-learning_rate*dw
        b = b-learning_rate*db

        # 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

4.5 预测

用已经训练好的参数w和b来预测数据集X,为其打上标签0或1 预测计算的主要步骤如下:

  • 计算
  • 计算结果若 <= 0.5,则打上标签0,计算结果若 > 0.5,则打上标签1;将预测结果存入向量Y_prediction
# 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
    A = sigmoid(np.dot(w.T, X)+b)

    for i in range(A.shape[1]):
        # Convert probabilities A[0,i] to actual predictions p[0,i]
        if A[0, i] > 0.5:
            Y_prediction[0, i] = 1
        else:
            Y_prediction[0, i] = 0

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

5 将所有的函数整合到同一个模型中

# 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.
    """

    # initialize parameters with zeros
    w = np.zeros((X_train.shape[0], 1))
    b = 0
    # Gradient descent
    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
    Y_prediction_test = predict(w, b, X_test)
    Y_prediction_train = predict(w, b, X_train)

    # 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

使用以下代码来测试模型:

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

预期输出:

Train Accuracy Test Accuracy
99.04306220095694 % 70.0 %

6 进一步分析

选择学习率 学习率α决定了更新参数的速度。如果学习率太大,我们可能会“超调”最佳值。同样,如果它太小,我们将需要经过较多次的迭代来收敛到最佳值。使用良好调整的学习率至关重要。

运行以下代码,查看不同的学习率会有怎样的影响:

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()

运行结果如下:

Learning Rate Train Accuracy Test Accuracy
0.01 99.52153110047847 % 68.0 %
0.001 88.99521531100478 % 64.0 %
0.0001 68.42105263157895 % 36.0 %

不同学习率下的成本函数曲线

注:

  • 不同的学习率产生不同的成本,因此有不同的预测结果
  • 较低的成本并不意味着更好的模型。必须检查是否有可能过度拟合。当训练精度远高于测试精度时,就会发生这种情况。
  • 在深度学习中,通常:
  • 选择更好地降低成本函数的学习率
  • 如果模型过度拟合,使用其他技术来减少过度拟合。

参考材料