TensorFlow 线性回归梯度下降模拟手写并可视化

python手写模拟梯度下降

以2元线性回归为例实现分类器：

            

线性回归函数：

误差函数（损失函数）：

每次梯度下降参数的变化：

用python模拟梯度下降算法，并进行鸢尾花数据集的分类

激励函数：

最终得到的线性回归方程的图像大致是这样（不要管图中数据，只是示意一下大致的模样）：

import matplotlib.pyplot as pltimport numpy as npfrom mpl_toolkits.mplot3d import Axes3Dfig = plt.figure()ax = Axes3D(fig)x = np.arange(1, 10, 0.02)y = np.arange(1, 10, 0.02)X, Y = np.meshgrid(x, y)Z = 1 + 2 * X - 3 * Yax.plot_surface(X, Y, Z, color='red')ax.contourf(X, Y, Z, zdir='z', offset=0, alpha=0.4)plt.show()

根据回归方程，分别划分为值为>0和<=0的两部分 

import numpy as npimport pandas as pdimport matplotlib.pyplot as pltfrom matplotlib.colors import ListedColormapclass AdalineGD:    def __init__(self, eta=0.01, n_iter=50):        self.eta = eta        self.n_iter = n_iter    def fit(self, X, y):        self.w_ = np.zeros(X.shape[1]+1)        self.J_ = []        for _ in range(self.n_iter):            yx = np.dot(X, self.w_[1:]) + self.w_[0]            self.w_[0] -= self.eta * (yx - y).sum()            self.w_[1:] -= self.eta * np.dot(X.T, (yx - y))            self.J_.append(((y-yx)**2).sum()*0.5)        return self    def predict(self, X):        return np.where(np.dot(X, self.w_[1:]) + self.w_[0] >= 0.0, 1, -1)def plot_decision_regions(X, y, classifier, ax):    # setup marker generator and color map    markers = ('x', 'o')    colors = ('red', 'blue')    cmap = ListedColormap(colors[:len(np.unique(y))])    # plot the decision region    x1_min, x1_max = X[:, 0].min() - 1, X[:, 0].max() + 1    x2_min, x2_max = X[:, 1].min() - 1, X[:, 1].max() + 1    xx1, xx2 = np.meshgrid(np.arange(x1_min, x1_max, 0.02), np.arange(x2_min, x2_max, 0.02))    z = classifier.predict(np.array([xx1.ravel(), xx2.ravel()]).T)    z = z.reshape(xx1.shape)    plt.contourf(xx1, xx2, z, cmap=cmap, apha=0.3)    plt.xlim(x1_min, x1_max)    plt.ylim(x2_min, x2_max)    # plot class samples    for idx, cl in enumerate(np.unique(y)):        ax[1].scatter(x=X[y == cl, 0], y=X[y == cl, 1], cmap=cmap(idx), marker=markers[idx], label=cl, alpha=1)    ax[1].set_title('Adaline-梯度下降法', fontproperties='SimHei')    ax[1].set_xlabel('经标准化处理的萼片宽度', fontproperties='SimHei')    ax[1].set_ylabel('经标准化处理的花瓣宽度', fontproperties='SimHei')df = pd.read_csv(r'http://archive.ics.uci.edu/ml/machine-learning-databases/iris/iris.data')y = df.iloc[:100, 4].valuesy = np.where(y == 'Iris-setosa', -1, 1)X = df.iloc[:100, [0, 2]].valuesX_std = X.copy()X_std[:, 0] = (X[:, 0] - X[:, 0].mean()) / X[:, 0].std()X_std[:, 1] = (X[:, 1] - X[:, 1].mean()) / X[:, 1].std()fig1, ax1 = plt.subplots(1, 2, figsize=(8, 4))demo = AdalineGD(n_iter=20, eta=0.01).fit(X_std, y)plot_decision_regions(X_std, y, demo, ax1)ax1[0].plot(range(1, len(demo.J_)+1), np.log10(demo.J_), marker='o')ax1[0].set_title('学习速率0.01', fontproperties='SimHei')ax1[0].set_xlabel('迭代次数', fontproperties='SimHei')ax1[0].set_ylabel('log(误差J)', fontproperties='SimHei')plt.show()

使用TensorFlow框架

import tensorflow as tfimport numpy as npimport matplotlib.pyplot as pltdef add_layer(input, in_size, out_size, activation_function=None):    Weight = tf.Variable(tf.random_normal([in_size, out_size]))    biases = tf.Variable(tf.zeros([1, out_size]))    y = tf.matmul(input, Weight) + biases    if activation_function is None:        return y    else:        return activation_function(y)X_data = np.linspace(-1, 1, 100, dtype=np.float32)[:, np.newaxis]noise = np.random.normal(0, 0.05, (X_data.shape[0], 1))# 使得产生的数据在x^2+0.5曲线上下y_data = np.square(X_data) + 0.5 + noiseX = tf.placeholder(tf.float32, [None, 1])y = tf.placeholder(tf.float32, [None, 1])# 通过add_layer指定了该层框架，之后在迭代过程中不再调用函数# 输入层为1个神经元，隐藏层为10个神经元，输出层为1个神经元hidden_layer = add_layer(X, 1, 10, activation_function=tf.nn.relu)output_layer = add_layer(hidden_layer, 10, 1, activation_function=None)loss = tf.reduce_mean(tf.square(y - output_layer))trainer = tf.train.GradientDescentOptimizer(0.1).minimize(loss)fig, ax = plt.subplots(1, 1)ax.scatter(X_data, y_data)with tf.Session() as sess:    sess.run(tf.global_variables_initializer())    for _ in range(301):        sess.run(trainer, feed_dict={X: X_data, y: y_data})        if _ % 50 == 0:            print(sess.run(loss, feed_dict={X: X_data, y: y_data}))            curve = ax.plot(X_data, sess.run(output_layer, feed_dict={X: X_data, y: y_data}))            plt.pause(0.5)  # 停留0.5s            if _ != 300:                ax.lines.remove(curve[0])  # 抹除ax上的线，必须以列表下标的形式plt.show()

线性回归，梯度下降算法可视化：

import tensorflow as tfimport numpy as npimport matplotlib.pyplot as pltfrom mpl_toolkits.mplot3d import Axes3Dlr = 0.1real_params = [1.2, 2.5]   # 真正的参数tf_X = tf.placeholder(tf.float32, [None, 1])tf_y = tf.placeholder(tf.float32, [None, 1])weight = tf.Variable(initial_value=[[5]], dtype=tf.float32)bia = tf.Variable(initial_value=[[4]], dtype=tf.float32)y = tf.matmul(tf_X, weight) + bialoss = tf.losses.mean_squared_error(tf_y, y)train_op = tf.train.GradientDescentOptimizer(lr).minimize(loss)X_data = np.linspace(-1, 1, 200)[:, np.newaxis]noise = np.random.normal(0, 0.1, X_data.shape)y_data = X_data * real_params[0] + real_params[1] + noisesess = tf.Session()sess.run(tf.global_variables_initializer())weights = []biases = []losses = []for step in range(400):    w, b, cost, _ = sess.run([weight, bia, loss, train_op],                             feed_dict={tf_X: X_data, tf_y: y_data})    weights.append(w)    biases.append(b)    losses.append(cost)result = sess.run(y, feed_dict={tf_X: X_data, tf_y: y_data})plt.figure(1)plt.scatter(X_data, y_data, color='r', alpha=0.5)plt.plot(X_data, result, lw=3)fig = plt.figure(2)ax_3d = Axes3D(fig)w_3d, b_3d = np.meshgrid(np.linspace(-2, 7, 30), np.linspace(-2, 7, 30))loss_3d = np.array(    [np.mean(np.square((X_data * w_ + b_) - y_data))     for w_, b_ in zip(w_3d.ravel(), b_3d.ravel())]).reshape(w_3d.shape)ax_3d.plot_surface(w_3d, b_3d, loss_3d, cmap=plt.get_cmap('rainbow'))weights = np.array(weights).ravel()biases = np.array(biases).ravel()# 描绘初始点ax_3d.scatter(weights[0], biases[0], losses[0], s=30, color='r')ax_3d.set_xlabel('w')ax_3d.set_ylabel('b')ax_3d.plot(weights, biases, losses, lw=3, c='r')plt.show()

拟合线性函数：y=1.2 x + 2.5

设初始的参数w=5,b=4,lr=0.1的拟合图像和梯度下降图像：

更改学习速率lr=1.0的图像：