6-7 代码实现随机梯度下降

import numpy as np
from sklearn.metrics import r2_score

class LinearRegression:

    def __init__(self):
        """初始化Linear Regression模型"""
        self.coef_ = None
        self.interception_ = 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.interception_ = 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(X_b)
            except:
                return float('inf')

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

        def gradient_descent(X_b, y, initial_theta, eta, n_iters = 1e4, epsilon=1e-8):
            theta = initial_theta
            i_iter = 0
            while i_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
                i_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)
        self.interception_ = self._theta[0]
        self.coef_ = self._theta[1:]
        return self

    def fit_sgd(self, X_train, y_train, n_iters = 5, t0 = 5, t1 = 50):
        """根据训练数据集X_train, y_train，使用随机梯度下降法训练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 dJ_sgd(theta, X_b_i, y_i):
            return X_b_i.T.dot(X_b_i.dot(theta)-y_i) * 2.

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

        def sgd(X_b, y, initial_theta, n_iters, t0, t1): # n_iters:对所有的样本看几圈
            theta = initial_theta
            m = len(X_b)

            i_iter = 0
            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])
                    last_theta = theta
                    theta = theta - learning_rate(i_iter*m+i, t0, t1) * gradient
            return theta

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

测试数据 + sgd

m = 100000

x = np.random.normal(size=m)
X = x.reshape(-1, 1)
y = 4. * x + 3. + np.random.normal(0, 3, size=m)

lin_reg = LinearRegression()
lin_reg.fit_sgd(X, y, n_iters=2)

刚开始在代码中犯了个错误，没有把L78的i_iter改成i_iter*m+i， 导致每次训练得到的模型差点都非常大，且偏离正确值也非常大。 改掉之后就好了， 可以如果学习率使用固定值，不能得到很好的效果。

真实数据 + sgd

真实数据

from sklearn import datasets

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

X = X[y<50.0]
y = y[y<50.0]

预处理

from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=666)

from sklearn.preprocessing import StandardScaler
standScaler = StandardScaler()
standScaler.fit(X_train)
X_train_standard = standScaler.transform(X_train)
X_test_standard = standScaler.transform(X_test)

SGD

lin_reg = LinearRegression()
%time lin_reg.fit_sgd(X_train_standard, y_train)
lin_reg.score(X_test_standard, y_test)

n_iters对score和Wall time的影响

n_iters	score	Wall time
5	0.7763594773981595	30.7 ms
50	0.8130771495096732	271 ms
100	0.8131205440883096	462 ms

真实数据 + sklearn的SGD

from sklearn.linear_model import SGDRegressor

sgd_lin = SGDRegressor()
%time sgd_lin.fit(X_train_standard, y_train)
sgd_lin.score(X_test_standard, y_test)

模型的得分差不多，但sklearn的SGD的速度明显快很多。 因为sklearn的SGD的实现过程与课程中有很大的不同。

视频还测试了SGDRegressor的n_iter参数。 这个参数在我用的sklearn版本中已经没有了。