梯度下降法-4.向量化和数据标准化

r囧r小猫 2023-08-17 16:44 94阅读 0赞

## 梯度下降的向量化 ##

\\\[\\Lambda J = \\begin\{bmatrix\} \\frac\{\\partial J\}\{\\partial \\theta \_0\} \\\\ \\frac\{\\partial J\}\{\\partial \\theta \_1\} \\\\ \\frac\{\\partial J\}\{\\partial \\theta \_2\} \\\\ ... \\\\ \\frac\{\\partial J\}\{\\partial \\theta \_n\} \\end\{bmatrix\} = \\frac\{2\}\{m\}\\begin\{bmatrix\} \\sum(X^\{(i)\}\_b\\theta - y^\{(i)\}) \\\\ \\sum(X^\{(i)\}\_b\\theta - y^\{(i)\}) \\cdot X\_1^\{(i)\}\\\\ \\sum(X^\{(i)\}\_b\\theta - y^\{(i)\}) \\cdot X\_2^\{(i)\}\\\\ ...\\\\ \\sum(X^\{(i)\}\_b\\theta - y^\{(i)\}) \\cdot X\_n^\{(i)\} \\end\{bmatrix\}\\\]  
之前求梯度的过程是使用for循环对每一项的\$\\theta\$ 求偏导数， 对上面的式子，进行向量化  
\\\[\\Lambda J = \\begin\{bmatrix\} \\frac\{\\partial J\}\{\\partial \\theta \_0\} \\\\ \\frac\{\\partial J\}\{\\partial \\theta \_1\} \\\\ \\frac\{\\partial J\}\{\\partial \\theta \_2\} \\\\ ... \\\\ \\frac\{\\partial J\}\{\\partial \\theta \_n\} \\end\{bmatrix\} = \\frac\{2\}\{m\}\\begin\{bmatrix\} \\sum(X^\{(i)\}\_b\\theta - y^\{(i)\}) \\cdot X\_0^\{(i)\}\\\\ \\sum(X^\{(i)\}\_b\\theta - y^\{(i)\}) \\cdot X\_1^\{(i)\}\\\\ \\sum(X^\{(i)\}\_b\\theta - y^\{(i)\}) \\cdot X\_2^\{(i)\}\\\\ ...\\\\ \\sum(X^\{(i)\}\_b\\theta - y^\{(i)\}) \\cdot X\_n^\{(i)\} \\end\{bmatrix\} = \\frac\{2\}\{m\}(X\_b\\theta -y)^T\\cdot X\_b \\\]  
将式子转换为列向量：  
\\\[\\Lambda J = \\frac\{2\}\{m\}(X\_b\\theta -y)^T\\cdot X\_b = \\frac\{2\}\{m\} X\_b^T \\cdot (X\_b\\theta -y)\\\]

优化代码：

def dJ(theta,X_b,y):  #求导
    
        #返回的导数矩阵    
        #res = numpy.empty(len(theta))
        #res[0] = numpy.sum(X_b.dot(theta)-y)
        #for i in range(1,len(theta)):
        #res[i] = ((X_b.dot(theta)-y).dot(X_b[:,i]))
        #return res*2/len(X_b)
    
        return X_b.T.dot(X_b.dot(theta)-y)*2/len(X_b)

使用梯度下降法预测波士顿房价：

import numpy
    import matplotlib.pyplot as plt
    from sklearn import datasets
    from mylib.LineRegression import LineRegression
    from mylib.model_selection import train_test_split
    
    boston = datasets.load_boston()
    X = boston.data
    y = boston.target
    X = X[y<50]
    y = y[y<50]
    
    X_train,X_test,y_train,y_test = train_test_split(X,y,seed=666)
    
    reg = LineRegression()
    %time reg.fit_gd(X_train,y_train)

![1465169-20190717211946614-957983338.png][]  
由此看出是梯度下降过程不收敛，先尝试减小学习步长 $\\eta $ 并增大学习次数n\_iters

%time reg.fit_gd(X_train,y_train,eta=0.000001,n_iters=1e6)
    reg.score(X_test,y_test)

![1465169-20190717212003826-106302603.png][]  
训练时间长达8分钟！！！

## 数据标准化 ##

由上可知，因为真实数据集各个特征的规模不一样，学习率取较大值时，梯度下降过程可能不收敛，学习率取较小值时，学习次数多，导致学习时间很慢，所以在使用梯度下降法时，将数据归一化  
![1465169-20190717212027866-754145554.png][]

from sklearn.preprocessing import StandardScaler
    Sd = StandardScaler()
    
    Sd.fit(X_train)
    X_train_stand = Sd.transform(X_train)
    X_test_stand = Sd.transform(X_test)
    
    %time reg.fit_gd(X_train_stand,y_train)

![1465169-20190717212042318-1011768490.png][]

### 梯度下降法的优势 ###

虚构一个有1000个样本数，每个样本有5000个特征的数据集

import numpy
    
    m=1000 #样本数
    n=5000 #特征值
    
    big_x = numpy.random.normal(size=(m,n))
    true_theta = numpy.random.uniform(0.0, 100.0,size=n+1)
    big_y = big_x.dot(true_theta[1:]) + true_theta[:1] + numpy.random.normal(0.0,10.0,size=m)

用线性回归算法训练：  
![1465169-20190717212059229-971217714.png][]

用梯度下降法训练：  
![1465169-20190717212111160-704741074.png][]

由上可以看出，梯度下降所需的时间要少于线性回归法，这是因为正规方程解处理的是m\*n规模的矩阵之间的大量乘法运算，当数据规模比较大的时候，计算耗时是更高的。

转载于:https://www.cnblogs.com/shuai-long/p/11203757.html

[1465169-20190717211946614-957983338.png]: /images/20230809/2d822d6eb1e2403ab1d32800bae19a13.png
[1465169-20190717212003826-106302603.png]: /images/20230809/8de92b8dba2f418ab6afec22168831ab.png
[1465169-20190717212027866-754145554.png]: /images/20230809/3b3689b53b8b4b7f9b5c15e9cda75759.png
[1465169-20190717212042318-1011768490.png]: /images/20230809/b552b6694c0c4ce4899144d2976400d1.png
[1465169-20190717212059229-971217714.png]: /images/20230809/bfdde1fa281f4099bf40417155e37bc6.png
[1465169-20190717212111160-704741074.png]: /images/20230809/00ec65499a354903b33da4e43c4cfbf2.png