一、梯度下降
梯度下降.png
梯度下降2.png
梯度下降大家族.png
二、代码的实现
（一.梯度下降）
导包
import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression
构建数据
X = np.linspace(-2,12,40).reshape(-1,1)
w = np.random.randint(1,9,size = 1)
b = np.random.randint(-5,5,size = 1)
# 增加了噪声
y = w*X + b + np.random.randn(40,1)*2
plt.scatter(X,y)
<matplotlib.collections.PathCollection at 0x1d379fb2148>
output_3_1.png
梯度下降
# 没有使用矩阵，实数
class LinearModel(object):
def __init__(self):#初始化，随机给定斜率和截距
self.w = np.random.randn(1)[0]
self.b = np.random.randn(1)[0]
def model(self,x):# 模型
return self.w*x + self.b#一元一次线性方程，模型
def loss(self,x,y):#损失，最小二乘法
cost = (self.model(x) - y)**2 # 损失函数越小越好
# 求解梯度，两个未知数，所以，偏导
d_w = 2*(self.model(x) - y)*x # 斜率w的偏导
d_b = 2*(self.model(x) - y)*1 # 截距b的偏导
return cost,d_w,d_b
def gradient_descent(self,step,d_w,d_b):# 梯度下降
self.w -= step*d_w # 根据梯度，更新斜率
self.b -= step*d_b # 根据梯度，更细截距
def fit(self,X,y):#fit 训练模型，将数据交给模型，寻找规律
precision = 1e-4 # 精确度
last_w = self.w + 0.01
last_b = self.b + 0.01
while True:
if (np.abs(self.w - last_w) < precision) & (np.abs(self.b - last_b) < precision):
break
# 更新斜率和截距
last_w = self.w # 更新之前，先保留，记录
last_b = self.b
cost_ = 0
dw_ = 0
db_ = 0
for i in range(40):#计算40个，返回40个偏导数，求平均值
cost,dw,db = self.loss(X[i,0],y[i,0])
cost_ += cost/40
dw_ += dw/40
db_ += db/40
self.gradient_descent(0.02,dw_,db_)
print('----------------------',self.w,self.b)
def predict(self,X):
return self.model(X)
使用梯度下降，可视化
X_test = np.linspace(-2,12,512).reshape(-1,1)
linear = LinearModel()
linear.fit(X,y)
y_ = linear.predict(X_test)
plt.plot(X_test,y_,color = 'green')
plt.scatter(X,y,color = 'red')
---------------------- 5.022756299270281 -0.4775691359043576
<matplotlib.collections.PathCollection at 0x1d37b20d488>
output_7_2.png
(二。梯度下降矩阵)
import numpy as np
import matplotlib.pyplot as plt
X = np.linspace(-2,12,40).reshape(-1,1)
w = np.random.randint(2,12,size = 1)
b = np.random.randint(-10,10,size = 1)
y = X*w + b + np.random.randn(40,1)*2.5
print(y.shape)
display(y)
# 将y.reshape(-1)一维的
y = y.reshape(-1)
print(y.shape)
display(y)
plt.scatter(X,y,color = 'red')
(40, 1)
array([[ 0.26477396],
[ 0.14625252],
[ 4.34337223],
[ 5.15737839],
[ 5.73058692],
[ 3.2005747 ],
[ 2.65238981],
[ 3.15718199],
[ 5.9824348 ],
[ 7.25627524],
[ 6.39046658],
[ 4.79229391],
[ 9.41034704],
[ 7.03066786],
[10.54726642],
[ 9.6647074 ],
[ 8.57165543],
[15.14317764],
[15.01679279],
[16.21608669],
[11.93008586],
[11.35315216],
[18.41892082],
[13.540176 ],
[13.21526135],
[16.17345117],
[15.47522667],
[17.14348976],
[23.00294753],
[18.58565509],
[19.42419585],
[21.30908848],
[20.62667937],
[26.9984373 ],
[23.69327916],
[25.71868151],
[28.15089499],
[25.08194538],
[31.16719531],
[30.02092197]])
(40,)
array([ 0.26477396, 0.14625252, 4.34337223, 5.15737839, 5.73058692,
3.2005747 , 2.65238981, 3.15718199, 5.9824348 , 7.25627524,
6.39046658, 4.79229391, 9.41034704, 7.03066786, 10.54726642,
9.6647074 , 8.57165543, 15.14317764, 15.01679279, 16.21608669,
11.93008586, 11.35315216, 18.41892082, 13.540176 , 13.21526135,
16.17345117, 15.47522667, 17.14348976, 23.00294753, 18.58565509,
19.42419585, 21.30908848, 20.62667937, 26.9984373 , 23.69327916,
25.71868151, 28.15089499, 25.08194538, 31.16719531, 30.02092197])
<matplotlib.collections.PathCollection at 0x1774afbbac8>
output_1_5.png
用方法，实现梯度下降
m是样本的数量
对数据X增加了一列，这一列对应着，截距
# 作为训练数据，增加了一列，截距
X_train = np.concatenate([X,np.ones(shape = (40,1))],axis = 1)
X_train
array([[-2. , 1. ],
[-1.64102564, 1. ],
[-1.28205128, 1. ],
[-0.92307692, 1. ],
[-0.56410256, 1. ],
[-0.20512821, 1. ],
[ 0.15384615, 1. ],
[ 0.51282051, 1. ],
[ 0.87179487, 1. ],
[ 1.23076923, 1. ],
[ 1.58974359, 1. ],
[ 1.94871795, 1. ],
[ 2.30769231, 1. ],
[ 2.66666667, 1. ],
[ 3.02564103, 1. ],
[ 3.38461538, 1. ],
[ 3.74358974, 1. ],
[ 4.1025641 , 1. ],
[ 4.46153846, 1. ],
[ 4.82051282, 1. ],
[ 5.17948718, 1. ],
[ 5.53846154, 1. ],
[ 5.8974359 , 1. ],
[ 6.25641026, 1. ],
[ 6.61538462, 1. ],
[ 6.97435897, 1. ],
[ 7.33333333, 1. ],
[ 7.69230769, 1. ],
[ 8.05128205, 1. ],
[ 8.41025641, 1. ],
[ 8.76923077, 1. ],
[ 9.12820513, 1. ],
[ 9.48717949, 1. ],
[ 9.84615385, 1. ],
[10.20512821, 1. ],
[10.56410256, 1. ],
[10.92307692, 1. ],
[11.28205128, 1. ],
[11.64102564, 1. ],
[12. , 1. ]])
根据矩阵求解的梯度，进行梯度下降
生成系数时，必须考虑形状
def gradient_descent(X,y):
theta = np.random.randn(2) # theta中既有斜率，又有截距
last_theta = theta + 0.1 #记录theta更新后，和上一步的误差
precision = 1e-4 #精确度
epsilon = 0.01 #步幅
while True:
# 当斜率和截距误差小于万分之一时，退出
if (np.abs(theta - last_theta) < precision).all():
break
# 更新
last_theta = theta.copy()
# 梯度下降，梯度是矩阵计算返回的
theta -= epsilon*2/40*X.T.dot(X.dot(theta) - y)
return theta
w_,b_ = gradient_descent(X_train,y)
j = lambda x : w_*x + b_
plt.scatter(X[:,0],y,color = 'red')
x_test = np.linspace(-2,12,1024)
y_ = j(x_test)
plt.plot(x_test,y_,color = 'green')
[<matplotlib.lines.Line2D at 0x1774b10ff08>]
output_10_1.png
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