机器学习-梯度下降法

不是一个机器学习算法
是一种基于搜索的最优化方法
作用：最小化一个损失函数
梯度上升法：最大化一个效用函数

并不是所有函数都有唯一的极值点

解决方法：

多次运行，随机化初始点
梯度下降法的初始点也是一个超参数

代码演示

python 复制代码

import numpy as np
import matplotlib.pyplot as plt
plot_x = np.linspace(-1., 6., 141)
plot_y = (plot_x-2.5)**2 - 1.
plt.plot(plot_x, plot_y)
plt.show()

梯度下降法

python 复制代码

epsilon = 1e-8
eta = 0.1
def J(theta):
    return (theta-2.5)**2 - 1.

def dJ(theta):
    return 2*(theta-2.5)

theta = 0.0
while True:
    gradient = dJ(theta)
    last_theta = theta
    theta = theta - eta * gradient
    
    if(abs(J(theta) - J(last_theta)) < epsilon):
        break
    
print(theta)
print(J(theta))

可视化

python 复制代码

theta = 0.0
theta_history = [theta]
while True:
    gradient = dJ(theta)
    last_theta = theta
    theta = theta - eta * gradient
    theta_history.append(theta)
    
    if(abs(J(theta) - J(last_theta)) < epsilon):
        break

plt.plot(plot_x, J(plot_x))
plt.plot(np.array(theta_history), J(np.array(theta_history)), color="r", marker='+')
plt.show()

封装

python 复制代码

def gradient_descent(initial_theta, eta, epsilon=1e-8):
    theta = initial_theta
    theta_history.append(initial_theta)

    while True:
        gradient = dJ(theta)
        last_theta = theta
        theta = theta - eta * gradient
        theta_history.append(theta)
    
        if(abs(J(theta) - J(last_theta)) < epsilon):
            break
            
def plot_theta_history():
    plt.plot(plot_x, J(plot_x))
    plt.plot(np.array(theta_history), J(np.array(theta_history)), color="r", marker='+')
    plt.show()

eta = 0.01时

python 复制代码

eta = 0.01
theta_history = []
gradient_descent(0, eta)
plot_theta_history()

eta = 0.001时

python 复制代码

eta = 0.001
theta_history = []
gradient_descent(0, eta)
plot_theta_history()

eta = 0.8时

python 复制代码

eta = 0.8
theta_history = []
gradient_descent(0, eta)
plot_theta_history()

优化避免死循环

python 复制代码

def J(theta):
    try:
        return (theta-2.5)**2 - 1.
    except:
        return float('inf')
def gradient_descent(initial_theta, eta, n_iters = 1e4, epsilon=1e-8):
    
    theta = initial_theta
    i_iter = 0
    theta_history.append(initial_theta)

    while i_iter < n_iters:
        gradient = dJ(theta)
        last_theta = theta
        theta = theta - eta * gradient
        theta_history.append(theta)
    
        if(abs(J(theta) - J(last_theta)) < epsilon):
            break
            
        i_iter += 1
        
    return

eta = 1.1时

python 复制代码

eta = 1.1
theta_history = []
gradient_descent(0, eta, n_iters=10)
plot_theta_history()

多元线性回归中的梯度下降法

代码

生成数据

python 复制代码

import numpy as np
import matplotlib.pyplot as plt
np.random.seed(666)
x = 2 * np.random.random(size=100)
y = x * 3. + 4. + np.random.normal(size=100)
X = x.reshape(-1, 1)
plt.scatter(x, y)
plt.show()

使用梯度下降法训练

python 复制代码

def J(theta, X_b, y):
    try:
        return np.sum((y - X_b.dot(theta))**2) / len(X_b)
    except:
        return float('inf')
        
def dJ(theta, X_b, y):
    res = np.empty(len(theta))
    res[0] = np.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)

def gradient_descent(X_b, y, initial_theta, eta, n_iters = 1e4, epsilon=1e-8):
    
    theta = initial_theta
    cur_iter = 0

    while cur_iter < n_iters:
        gradient = dJ(theta, X_b, y)
        last_theta = theta
        theta = theta - eta * gradient
        if(abs(J(theta, X_b, y) - J(last_theta, X_b, y)) < epsilon):
            break
            
        cur_iter += 1

    return theta

python 复制代码

X_b = np.hstack([np.ones((len(x), 1)), x.reshape(-1,1)])
initial_theta = np.zeros(X_b.shape[1])
eta = 0.01

theta = gradient_descent(X_b, y, initial_theta, eta)

封装

python 复制代码

    def fit_gd(self, X_train, y_train, eta=0.01, n_iters=1e4):
        """根据训练数据集X_train, y_train, 使用梯度下降法训练Linear Regression模型"""
        assert X_train.shape[0] == y_train.shape[0], \
            "the size of X_train must be equal to the size of y_train"

        def J(theta, X_b, y):
            try:
                return np.sum((y - X_b.dot(theta)) ** 2) / len(y)
            except:
                return float('inf')

        def dJ(theta, X_b, y):
            res = np.empty(len(theta))
            res[0] = np.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)

        def gradient_descent(X_b, y, initial_theta, eta, n_iters=1e4, epsilon=1e-8):

            theta = initial_theta
            cur_iter = 0

            while cur_iter < n_iters:
                gradient = dJ(theta, X_b, y)
                last_theta = theta
                theta = theta - eta * gradient
                if (abs(J(theta, X_b, y) - J(last_theta, X_b, y)) < epsilon):
                    break

                cur_iter += 1

            return theta

        X_b = np.hstack([np.ones((len(X_train), 1)), X_train])
        initial_theta = np.zeros(X_b.shape[1])
        self._theta = gradient_descent(X_b, y_train, initial_theta, eta, n_iters)

        self.intercept_ = self._theta[0]
        self.coef_ = self._theta[1:]

        return self

全：

python 复制代码

import numpy as np
from .metrics import r2_score

class LinearRegression:

    def __init__(self):
        """初始化Linear Regression模型"""
        self.coef_ = None
        self.intercept_ = None
        self._theta = None

    def fit_normal(self, X_train, y_train):
        """根据训练数据集X_train, y_train训练Linear Regression模型"""
        assert X_train.shape[0] == y_train.shape[0], \
            "the size of X_train must be equal to the size of y_train"

        X_b = np.hstack([np.ones((len(X_train), 1)), X_train])
        self._theta = np.linalg.inv(X_b.T.dot(X_b)).dot(X_b.T).dot(y_train)

        self.intercept_ = self._theta[0]
        self.coef_ = self._theta[1:]

        return self

    def fit_gd(self, X_train, y_train, eta=0.01, n_iters=1e4):
        """根据训练数据集X_train, y_train, 使用梯度下降法训练Linear Regression模型"""
        assert X_train.shape[0] == y_train.shape[0], \
            "the size of X_train must be equal to the size of y_train"

        def J(theta, X_b, y):
            try:
                return np.sum((y - X_b.dot(theta)) ** 2) / len(y)
            except:
                return float('inf')

        def dJ(theta, X_b, y):
            res = np.empty(len(theta))
            res[0] = np.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)

        def gradient_descent(X_b, y, initial_theta, eta, n_iters=1e4, epsilon=1e-8):

            theta = initial_theta
            cur_iter = 0

            while cur_iter < n_iters:
                gradient = dJ(theta, X_b, y)
                last_theta = theta
                theta = theta - eta * gradient
                if (abs(J(theta, X_b, y) - J(last_theta, X_b, y)) < epsilon):
                    break

                cur_iter += 1

            return theta

        X_b = np.hstack([np.ones((len(X_train), 1)), X_train])
        initial_theta = np.zeros(X_b.shape[1])
        self._theta = gradient_descent(X_b, y_train, initial_theta, eta, n_iters)

        self.intercept_ = self._theta[0]
        self.coef_ = self._theta[1:]

        return self

    def predict(self, X_predict):
        """给定待预测数据集X_predict，返回表示X_predict的结果向量"""
        assert self.intercept_ is not None and self.coef_ is not None, \
            "must fit before predict!"
        assert X_predict.shape[1] == len(self.coef_), \
            "the feature number of X_predict must be equal to X_train"

        X_b = np.hstack([np.ones((len(X_predict), 1)), X_predict])
        return X_b.dot(self._theta)

    def score(self, X_test, y_test):
        """根据测试数据集 X_test 和 y_test 确定当前模型的准确度"""

        y_predict = self.predict(X_test)
        return r2_score(y_test, y_predict)

    def __repr__(self):
        return "LinearRegression()"

线性回归中使用梯度下降法

优化代码

python 复制代码

  def dJ(theta, X_b, y):
            return X_b.T.dot(X_b.dot(theta) - y) * 2. / len(y)

python 复制代码

import numpy as np
from .metrics import r2_score

class LinearRegression:

    def __init__(self):
        """初始化Linear Regression模型"""
        self.coef_ = None
        self.intercept_ = None
        self._theta = None

    def fit_normal(self, X_train, y_train):
        """根据训练数据集X_train, y_train训练Linear Regression模型"""
        assert X_train.shape[0] == y_train.shape[0], \
            "the size of X_train must be equal to the size of y_train"

        X_b = np.hstack([np.ones((len(X_train), 1)), X_train])
        self._theta = np.linalg.inv(X_b.T.dot(X_b)).dot(X_b.T).dot(y_train)

        self.intercept_ = self._theta[0]
        self.coef_ = self._theta[1:]

        return self

    def fit_gd(self, X_train, y_train, eta=0.01, n_iters=1e4):
        """根据训练数据集X_train, y_train, 使用梯度下降法训练Linear Regression模型"""
        assert X_train.shape[0] == y_train.shape[0], \
            "the size of X_train must be equal to the size of y_train"

        def J(theta, X_b, y):
            try:
                return np.sum((y - X_b.dot(theta)) ** 2) / len(y)
            except:
                return float('inf')
            
        def dJ(theta, X_b, y):
            return X_b.T.dot(X_b.dot(theta) - y) * 2. / len(y)

        def gradient_descent(X_b, y, initial_theta, eta, n_iters=1e4, epsilon=1e-8):

            theta = initial_theta
            cur_iter = 0

            while cur_iter < n_iters:
                gradient = dJ(theta, X_b, y)
                last_theta = theta
                theta = theta - eta * gradient
                if (abs(J(theta, X_b, y) - J(last_theta, X_b, y)) < epsilon):
                    break

                cur_iter += 1

            return theta

        X_b = np.hstack([np.ones((len(X_train), 1)), X_train])
        initial_theta = np.zeros(X_b.shape[1])
        self._theta = gradient_descent(X_b, y_train, initial_theta, eta, n_iters)

        self.intercept_ = self._theta[0]
        self.coef_ = self._theta[1:]

        return self

    def predict(self, X_predict):
        """给定待预测数据集X_predict，返回表示X_predict的结果向量"""
        assert self.intercept_ is not None and self.coef_ is not None, \
            "must fit before predict!"
        assert X_predict.shape[1] == len(self.coef_), \
            "the feature number of X_predict must be equal to X_train"

        X_b = np.hstack([np.ones((len(X_predict), 1)), X_predict])
        return X_b.dot(self._theta)

    def score(self, X_test, y_test):
        """根据测试数据集 X_test 和 y_test 确定当前模型的准确度"""

        y_predict = self.predict(X_test)
        return r2_score(y_test, y_predict)

    def __repr__(self):
        return "LinearRegression()"

代码测试

python 复制代码

import numpy as np
from sklearn import datasets
boston = datasets.load_boston()
X = boston.data
y = boston.target

X = X[y < 50.0]
y = y[y < 50.0]
from playML.model_selection import train_test_split

X_train, X_test, y_train, y_test = train_test_split(X, y, seed=666)
from playML.LinearRegression import LinearRegression

lin_reg1 = LinearRegression()
%time lin_reg1.fit_normal(X_train, y_train)
lin_reg1.score(X_test, y_test)

使用梯度下降法

python 复制代码

lin_reg2 = LinearRegression()
lin_reg2.fit_gd(X_train, y_train)

更改eta值

python 复制代码

lin_reg2.fit_gd(X_train, y_train, eta=0.000001)
lin_reg2.score(X_test, y_test)

再优化

python 复制代码

%time lin_reg2.fit_gd(X_train, y_train, eta=0.000001, n_iters=1e6)
lin_reg2.score(X_test, y_test)

归一化

python 复制代码

from sklearn.preprocessing import StandardScaler

standardScaler = StandardScaler()
standardScaler.fit(X_train)
X_train_standard = standardScaler.transform(X_train)

lin_reg3 = LinearRegression()
%time lin_reg3.fit_gd(X_train_standard, y_train)
X_test_standard = standardScaler.transform(X_test)
lin_reg3.score(X_test_standard, y_test)

随机梯度下降法 Stochastic Gradient Descent

模拟退化的思想

代码

批量梯度下降法

python 复制代码

import numpy as np
import matplotlib.pyplot as plt
m = 100000

x = np.random.normal(size=m)
X = x.reshape(-1,1)
y = 4.*x + 3. + np.random.normal(0, 3, size=m)
def J(theta, X_b, y):
    try:
        return np.sum((y - X_b.dot(theta)) ** 2) / len(y)
    except:
        return float('inf')
    
def dJ(theta, X_b, y):
    return X_b.T.dot(X_b.dot(theta) - y) * 2. / len(y)

def gradient_descent(X_b, y, initial_theta, eta, n_iters=1e4, epsilon=1e-8):

    theta = initial_theta
    cur_iter = 0

    while cur_iter < n_iters:
        gradient = dJ(theta, X_b, y)
        last_theta = theta
        theta = theta - eta * gradient
        if (abs(J(theta, X_b, y) - J(last_theta, X_b, y)) < epsilon):
            break

        cur_iter += 1

    return theta
X_b = np.hstack([np.ones((len(X), 1)), X])
initial_theta = np.zeros(X_b.shape[1])
eta = 0.01
theta = gradient_descent(X_b, y, initial_theta, eta)

随机梯度下降法

python 复制代码

def dJ_sgd(theta, X_b_i, y_i):
    return 2 * X_b_i.T.dot(X_b_i.dot(theta) - y_i)

def sgd(X_b, y, initial_theta, n_iters):

    t0, t1 = 5, 50
    def learning_rate(t):
        return t0 / (t + t1)

    theta = initial_theta
    for cur_iter in range(n_iters):
        rand_i = np.random.randint(len(X_b))
        gradient = dJ_sgd(theta, X_b[rand_i], y[rand_i])
        theta = theta - learning_rate(cur_iter) * gradient

    return theta

python 复制代码

X_b = np.hstack([np.ones((len(X), 1)), X])
initial_theta = np.zeros(X_b.shape[1])
theta = sgd(X_b, y, initial_theta, n_iters=m//3)

封装

python 复制代码

def fit_sgd(self, X_train, y_train, n_iters=50, t0=5, t1=50):
        """根据训练数据集X_train, y_train, 使用梯度下降法训练Linear Regression模型"""
        assert X_train.shape[0] == y_train.shape[0], \
            "the size of X_train must be equal to the size of y_train"
        assert n_iters >= 1

        def dJ_sgd(theta, X_b_i, y_i):
            return X_b_i * (X_b_i.dot(theta) - y_i) * 2.

        def sgd(X_b, y, initial_theta, n_iters=5, t0=5, t1=50):

            def learning_rate(t):
                return t0 / (t + t1)

            theta = initial_theta
            m = len(X_b)
            for i_iter in range(n_iters):
                indexes = np.random.permutation(m)
                X_b_new = X_b[indexes,:]
                y_new = y[indexes]
                for i in range(m):
                    gradient = dJ_sgd(theta, X_b_new[i], y_new[i])
                    theta = theta - learning_rate(i_iter * m + i) * gradient

            return theta

python 复制代码

import numpy as np
from .metrics import r2_score

class LinearRegression:

    def __init__(self):
        """初始化Linear Regression模型"""
        self.coef_ = None
        self.intercept_ = None
        self._theta = None

    def fit_normal(self, X_train, y_train):
        """根据训练数据集X_train, y_train训练Linear Regression模型"""
        assert X_train.shape[0] == y_train.shape[0], \
            "the size of X_train must be equal to the size of y_train"

        X_b = np.hstack([np.ones((len(X_train), 1)), X_train])
        self._theta = np.linalg.inv(X_b.T.dot(X_b)).dot(X_b.T).dot(y_train)

        self.intercept_ = self._theta[0]
        self.coef_ = self._theta[1:]

        return self

    def fit_bgd(self, X_train, y_train, eta=0.01, n_iters=1e4):
        """根据训练数据集X_train, y_train, 使用梯度下降法训练Linear Regression模型"""
        assert X_train.shape[0] == y_train.shape[0], \
            "the size of X_train must be equal to the size of y_train"

        def J(theta, X_b, y):
            try:
                return np.sum((y - X_b.dot(theta)) ** 2) / len(y)
            except:
                return float('inf')

        def dJ(theta, X_b, y):
            return X_b.T.dot(X_b.dot(theta) - y) * 2. / len(y)

        def gradient_descent(X_b, y, initial_theta, eta, n_iters=1e4, epsilon=1e-8):

            theta = initial_theta
            cur_iter = 0

            while cur_iter < n_iters:
                gradient = dJ(theta, X_b, y)
                last_theta = theta
                theta = theta - eta * gradient
                if (abs(J(theta, X_b, y) - J(last_theta, X_b, y)) < epsilon):
                    break

                cur_iter += 1

            return theta

        X_b = np.hstack([np.ones((len(X_train), 1)), X_train])
        initial_theta = np.zeros(X_b.shape[1])
        self._theta = gradient_descent(X_b, y_train, initial_theta, eta, n_iters)

        self.intercept_ = self._theta[0]
        self.coef_ = self._theta[1:]

        return self

    def fit_sgd(self, X_train, y_train, n_iters=50, t0=5, t1=50):
        """根据训练数据集X_train, y_train, 使用梯度下降法训练Linear Regression模型"""
        assert X_train.shape[0] == y_train.shape[0], \
            "the size of X_train must be equal to the size of y_train"
        assert n_iters >= 1

        def dJ_sgd(theta, X_b_i, y_i):
            return X_b_i * (X_b_i.dot(theta) - y_i) * 2.

        def sgd(X_b, y, initial_theta, n_iters=5, t0=5, t1=50):

            def learning_rate(t):
                return t0 / (t + t1)

            theta = initial_theta
            m = len(X_b)
            for i_iter in range(n_iters):
                indexes = np.random.permutation(m)
                X_b_new = X_b[indexes,:]
                y_new = y[indexes]
                for i in range(m):
                    gradient = dJ_sgd(theta, X_b_new[i], y_new[i])
                    theta = theta - learning_rate(i_iter * m + i) * gradient

            return theta

        X_b = np.hstack([np.ones((len(X_train), 1)), X_train])
        initial_theta = np.random.randn(X_b.shape[1])
        self._theta = sgd(X_b, y_train, initial_theta, n_iters, t0, t1)

        self.intercept_ = self._theta[0]
        self.coef_ = self._theta[1:]

        return self

    def predict(self, X_predict):
        """给定待预测数据集X_predict，返回表示X_predict的结果向量"""
        assert self.intercept_ is not None and self.coef_ is not None, \
            "must fit before predict!"
        assert X_predict.shape[1] == len(self.coef_), \
            "the feature number of X_predict must be equal to X_train"

        X_b = np.hstack([np.ones((len(X_predict), 1)), X_predict])
        return X_b.dot(self._theta)

    def score(self, X_test, y_test):
        """根据测试数据集 X_test 和 y_test 确定当前模型的准确度"""

        y_predict = self.predict(X_test)
        return r2_score(y_test, y_predict)

    def __repr__(self):
        return "LinearRegression()"

真实使用我们自己的SGD

python 复制代码

from sklearn import datasets

boston = datasets.load_boston()
X = boston.data
y = boston.target

X = X[y < 50.0]
y = y[y < 50.0]
from playML.model_selection import train_test_split

X_train, X_test, y_train, y_test = train_test_split(X, y, seed=666)
from sklearn.preprocessing import StandardScaler

standardScaler = StandardScaler()
standardScaler.fit(X_train)
X_train_standard = standardScaler.transform(X_train)
X_test_standard = standardScaler.transform(X_test)
from playML.LinearRegression import LinearRegression

lin_reg = LinearRegression()
lin_reg.fit_sgd(X_train_standard, y_train, n_iters=100)
lin_reg.score(X_test_standard, y_test)

scikit-learn中的SGD

python 复制代码

from sklearn.linear_model import SGDRegressor

sgd_reg = SGDRegressor()
sgd_reg.fit(X_train_standard, y_train)
sgd_reg.score(X_test_standard, y_test)

python 复制代码

sgd_reg = SGDRegressor(n_iter=50)
sgd_reg.fit(X_train_standard, y_train)
sgd_reg.score(X_test_standard, y_test)

关于梯度的调试

生成数据

python 复制代码

import numpy as np
import matplotlib.pyplot as plt
np.random.seed(666)
X = np.random.random(size=(1000, 10))

true_theta = np.arange(1, 12, dtype=float)
X_b = np.hstack([np.ones((len(X), 1)), X])
y = X_b.dot(true_theta) + np.random.normal(size=1000)

python 复制代码

def J(theta, X_b, y):
    try:
        return np.sum((y - X_b.dot(theta))**2) / len(X_b)
    except:
        return float('inf')

python 复制代码

def dJ_math(theta, X_b, y):
    return X_b.T.dot(X_b.dot(theta) - y) * 2. / len(y)

python 复制代码

def dJ_debug(theta, X_b, y, epsilon=0.01):
    res = np.empty(len(theta))
    for i in range(len(theta)):
        theta_1 = theta.copy()
        theta_1[i] += epsilon
        theta_2 = theta.copy()
        theta_2[i] -= epsilon
        res[i] = (J(theta_1, X_b, y) - J(theta_2, X_b, y)) / (2 * epsilon)
    return res

python 复制代码

def gradient_descent(dJ, X_b, y, initial_theta, eta, n_iters = 1e4, epsilon=1e-8):
    
    theta = initial_theta
    cur_iter = 0

    while cur_iter < n_iters:
        gradient = dJ(theta, X_b, y)
        last_theta = theta
        theta = theta - eta * gradient
        if(abs(J(theta, X_b, y) - J(last_theta, X_b, y)) < epsilon):
            break
            
        cur_iter += 1

    return theta

python 复制代码

X_b = np.hstack([np.ones((len(X), 1)), X])
initial_theta = np.zeros(X_b.shape[1])
eta = 0.01

%time theta = gradient_descent(dJ_debug, X_b, y, initial_theta, eta)
theta

python 复制代码

%time theta = gradient_descent(dJ_math, X_b, y, initial_theta, eta)
theta