神经网络与深度学习（邱锡鹏）编程练习 2 实验4 基函数回归（梯度下降法优化）

多项式基函数

梯度下降优化过程中，产生“梯度爆炸”，在第14轮运算溢出。

    def gradient(phi_grad, y, w_init, lr=0.001, step_num=16):  # lr 学习率; step_num 迭代次数
        w_train = w_init
        for i in range(step_num):
            print("循环次数：", i, "参数 w_train = ", w_train)
            grad = phi_grad.T.dot(phi_grad.dot(w_train) - y) * 2.0 / len(phi_grad)  # 计算梯度
            w_train = w_train - lr * grad  # 更新 w
        return w_train

    init_theta = np.zeros(phi.shape[1])
    w = gradient(phi, y_train, init_theta)

循环次数调整到13以内（超出运算溢出之前），可以观察图像。

测试集变为“直线0”的原因：纵轴数值太大，1.6*10的253次幂。 测试集y值在0-25区间，看上去就都近似为0了。

 

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高斯基函数

使用梯度下降法，效果还可以，但是不如最小二乘法效果好

    def gradient(phi_grad, y, w_init, lr=0.1, step_num=10000):  # lr 学习率; step_num 迭代次数
        w_train = w_init
        for i in range(step_num):
            # print("循环次数：", i, "参数 w_train = ", w_train)
            grad = phi_grad.T.dot(phi_grad.dot(w_train) - y) * 2.0 / len(phi_grad)  # 计算梯度
            w_train = w_train - lr * grad  # 更新 w
        return w_train

    init_theta = np.zeros(phi.shape[1])
    w = gradient(phi, y_train, init_theta)

 

多项式基函数源代码：

查看代码

import numpy as np
import matplotlib.pyplot as plt


def load_data(filename):  # 载入数据
    xys = []
    with open(filename, 'r') as f:
        for line in f:
            xys.append(map(float, line.strip().split()))
        xs, ys = zip(*xys)
        return np.asarray(xs), np.asarray(ys)


def multinomial_basis(x, feature_num=10):
    x = np.expand_dims(x, axis=1)  # shape(N, 1)
    feat = [x]
    for i in range(2, feature_num + 1):
        feat.append(x ** i)
    ret = np.concatenate(feat, axis=1)
    return ret


def main(x_train, y_train):  # 训练模型，并返回从x到y的映射。
    basis_func = multinomial_basis  # shape(N, 1)的函数
    phi0 = np.expand_dims(np.ones_like(x_train), axis=1)  # shape(N,1)大小的全1 array
    phi1 = basis_func(x_train)  # 将x_train的shape转换为(N, 1)
    phi = np.concatenate([phi0, phi1], axis=1)  # phi.shape=(300,2) phi是增广特征向量的转置
    print("phi shape = ", phi.shape)

    # 梯度下降法 优化w
    def gradient(phi_grad, y, w_init, lr=0.0001, step_num=11):  # lr 学习率; step_num 迭代次数
        w_train = w_init
        for i in range(step_num):
            print("循环次数：", i, "参数 w_train = ", w_train)
            grad = phi_grad.T.dot(phi_grad.dot(w_train) - y) * 2.0 / len(phi_grad)  # 计算梯度
            w_train = w_train - lr * grad  # 更新 w
        return w_train

    init_theta = np.zeros(phi.shape[1])
    w = gradient(phi, y_train, init_theta)
    print("参数 w = ", w)

    def f(x):
        phi0 = np.expand_dims(np.ones_like(x), axis=1)
        # print("参数 phi0 = ", phi0)
        phi1 = basis_func(x)
        # print("参数 phi1 = ", phi1)
        phi = np.concatenate([phi0, phi1], axis=1)
        # print("参数 phi = ", phi)
        y = np.dot(phi, w)  # 矩阵乘法
        # print("参数 y = ", y)
        return y

    return f


def evaluate(ys, ys_pred):  # 评估模型
    std = np.sqrt(np.mean(np.abs(ys - ys_pred) ** 2))
    return std


if __name__ == '__main__':  # 程序主入口（建议不要改动以下函数的接口）
    train_file = 'train.txt'
    test_file = 'test.txt'
    # 载入数据
    x_train, y_train = load_data(train_file)
    x_test, y_test = load_data(test_file)

    print("x_train shape:", x_train.shape)
    print("x_test shape:", x_test.shape)

    # 训练模型，返回一个函数f()使得 y = f(x)
    f = main(x_train, y_train)

    # y_train_pred = f(x_train)  # 训练集 预测值
    # std = evaluate(y_train, y_train_pred)  # 使用训练集评估模型
    # print('训练集 预测值与真实值的标准差：{:.1f}'.format(std))
    # print("参数 y_test = ", y_test)
    y_test_pred = f(x_test)  # 测试集 预测值
    # print("参数 y_test = ", y_test)
    # print("参数 y_test_pred = ", y_test_pred)
    # print("参数 y_test shape = ", y_test.shape)
    # print("参数 y_test_pred shape = ", y_test_pred.shape)
    # std = evaluate(y_test, y_test_pred)  # 使用测试集评估模型
    # print('测试集 预测值与真实值的标准差：{:.1f}'.format(std))

    # 显示结果
    # plt.plot(x_train, y_train, 'r.')  # 训练集

    plt.plot(x_test, y_test_pred, 'k.')  # 测试集 的 预测值
    plt.plot(x_test, y_test, 'b.')  # 测试集
    plt.xlabel('x')
    plt.ylabel('y')
    plt.title('Linear Regression')
    plt.legend(['test', 'pred'])
    plt.show()

高斯基函数源代码：

查看代码

import numpy as np
import matplotlib.pyplot as plt


def load_data(filename):  # 载入数据
    xys = []
    with open(filename, 'r') as f:
        for line in f:
            xys.append(map(float, line.strip().split()))
        xs, ys = zip(*xys)
        return np.asarray(xs), np.asarray(ys)


def gaussian_basis(x, feature_num=10):
    centers = np.linspace(0, 25, feature_num)
    width = 1.0 * (centers[1] - centers[0])
    x = np.expand_dims(x, axis=1)
    x = np.concatenate([x] * feature_num, axis=1)

    out = (x - centers) / width
    ret = np.exp(-0.5 * out ** 2)
    return ret


def main(x_train, y_train):  # 训练模型，并返回从x到y的映射。
    basis_func = gaussian_basis  # shape(N, 1)的函数
    phi0 = np.expand_dims(np.ones_like(x_train), axis=1)  # shape(N,1)大小的全1 array
    phi1 = basis_func(x_train)  # 将x_train的shape转换为(N, 1)
    phi = np.concatenate([phi0, phi1], axis=1)  # phi.shape=(300,2) phi是增广特征向量的转置
    print("phi shape = ", phi.shape)

    # 梯度下降法 优化w
    def gradient(phi_grad, y, w_init, lr=0.1, step_num=10000):  # lr 学习率; step_num 迭代次数
        w_train = w_init
        for i in range(step_num):
            # print("循环次数：", i, "参数 w_train = ", w_train)
            grad = phi_grad.T.dot(phi_grad.dot(w_train) - y) * 2.0 / len(phi_grad)  # 计算梯度
            w_train = w_train - lr * grad  # 更新 w
        return w_train

    init_theta = np.zeros(phi.shape[1])
    w = gradient(phi, y_train, init_theta)

    def f(x):
        phi0 = np.expand_dims(np.ones_like(x), axis=1)
        phi1 = basis_func(x)
        phi = np.concatenate([phi0, phi1], axis=1)
        y = np.dot(phi, w)  # 矩阵乘法
        return y

    return f


def evaluate(ys, ys_pred):  # 评估模型
    std = np.sqrt(np.mean(np.abs(ys - ys_pred) ** 2))
    return std


if __name__ == '__main__':  # 程序主入口（建议不要改动以下函数的接口）
    train_file = 'train.txt'
    test_file = 'test.txt'
    # 载入数据
    x_train, y_train = load_data(train_file)
    x_test, y_test = load_data(test_file)
    print("x_train shape:", x_train.shape)
    print("x_test shape:", x_test.shape)

    # 训练模型，返回一个函数f()使得 y = f(x)
    f = main(x_train, y_train)

    y_train_pred = f(x_train)  # 训练集 预测值
    std = evaluate(y_train, y_train_pred)  # 使用训练集评估模型
    print('训练集 预测值与真实值的标准差：{:.1f}'.format(std))

    y_test_pred = f(x_test)  # 测试集 预测值
    std = evaluate(y_test, y_test_pred)  # 使用测试集评估模型
    print('测试集 预测值与真实值的标准差：{:.1f}'.format(std))

    # 显示结果
    # plt.plot(x_train, y_train, 'r.')  # 训练集
    plt.plot(x_test, y_test, 'b.')  # 测试集
    plt.plot(x_test, y_test_pred, 'k.')  # 测试集 的 预测值
    plt.xlabel('x')
    plt.ylabel('y')
    plt.title('Linear Regression')
    plt.legend(['train', 'test', 'pred'])
    plt.show()