{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "### 一元线性回归的梯度下降\n", "\n", "下面使用梯度下降算法来求解前面的线性回归问题，我们知道，对一元线性回归，损失函数为：\n", "\n", "$$\n", "loss = J(a, b) = \\frac{1}{2m} \\sum_{i=1}^{m}(y_i - (ax_i + b))^2\n", "$$\n", "\n", "这是一个二元二次方程，我们对该方程求梯度，也就是分别对参数 a 和 b 求偏导，得到参数的更新公式：\n", "\n", "$$\n", "\\begin{align}\n", "a &:= a - \\eta \\frac{\\partial J(a, b)}{\\partial a} \\\\\n", "b &:= b - \\eta \\frac{\\partial J(a, b)}{\\partial b} \\\\\n", "\\end{align}\n", "$$\n", "\n", "将损失函数带入上式求导，得到：\n", "\n", "$$\n", "\\begin{align}\n", "a &:= a - \\eta \\frac{1}{m} \\sum_{i=1}^{m}(y_i - (ax_i + b))x_i \\\\\n", "b &:= b - \\eta \\frac{1}{m} \\sum_{i=1}^{m}(y_i - (ax_i + b))\n", "\\end{align}\n", "$$\n", "\n", "我们使用代码来实现这个过程，先来看一个最简单的例子，手工造几个散点，然后使用梯度下降法来求解线性回归。如下所示：" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "import numpy as np\n", "\n", "X = np.array([0,1,2])\n", "Y = np.array([0.9, 3.1, 5.1])\n", "\n", "plt.plot(X, Y, 'k.')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "很显然这是一个一元线性回归，根据上面的介绍，我们可以定义出损失函数和参数 a, b 的更新公式：" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "# 损失函数\n", "def loss(X, Y, a, b):\n", " s = 0\n", " for i in range(X.size):\n", " s += ((a*X[i] + b)-Y[i])**2\n", " return s * (1./2*Y.size)\n", "\n", "# 参数 a 的更新公式\n", "def update_a(X, Y, a, b, eta):\n", " s = 0\n", " for i in range(X.size):\n", " s += ((a*X[i] + b)-Y[i])*X[i]\n", " return a - eta * s * (1./Y.size)\n", "\n", "# 参数 b 的更新公式\n", "def update_b(X, Y, a, b, eta):\n", " s = 0\n", " for i in range(X.size):\n", " s += ((a*X[i] + b)-Y[i])\n", " return b - eta * s * (1./Y.size)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "我们选取参数的初始值 $(a, b) = (0, 0)$，并令学习率 $\\eta = 0.01$，然后进行第一次迭代，并计算损失函数的值和上次损失函数的值之间的差值（这个差值可以用于判断梯度下降过程是否已经收敛，如果两次损失函数的差值足够小，就认为收敛）：" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "a = 0.04433333333333332, b = 0.03033333333333333, loss = 52.07898433333334, diff = 2.566015666666658\n" ] } ], "source": [ "a = 0\n", "b = 0\n", "eta = 0.01\n", "\n", "a_new = update_a(X, Y, a, b, eta)\n", "b_new = update_b(X, Y, a, b, eta)\n", "loss_new = loss(X, Y, a_new, b_new)\n", "diff = abs(loss(X, Y, a, b) - loss(X, Y, a_new, b_new))\n", "print(\"a = {0}, b = {1}, loss = {2}, diff = {3}\".format(a_new, b_new, loss_new, diff))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "我们不妨画出迭代一次后的图像，很显然，这个差值还比较大，我们需要继续这个过程：" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "image/png": 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R8US35+mAuyR9zPZrWlwOu9v2N7o7UumOSzoeEUv/mnpciyHvZx+W9F8RMRsRFyQ9Iem3uzxTp/yv7d2S1Pp4soyd9Eq4r7r3tbRtLa57HouIL3Z7nk6IiM9FxN6ImNTin/EPI6Kvj8Qi4n8k/cL2r7ceOijp5S6O1Ak/l/Qh29taf88Pqs9PyC7zpKRPtW5/StI/lbGTQm9dVrar9H0t75L0SUk/tf1C67G/ab1NHPrLX0p6tHVQ8qqkP+3yPKWKiGdtPy7pOS1ePfW8+vBVlLYfk1STtMv2cUmfl/QFSd+y/Wda/B/Yn5Syb145CQC59MpSCQCgIMINAMkQbgBIhnADQDKEGwCSIdwAkAzhBoBkCDcAJPP/RhLBCG6lxZUAAAAASUVORK5CYII=\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.plot(X, Y, 'k.')\n", "\n", "x = np.linspace(0, 10, 10)\n", "y = a_new*x+b_new\n", "\n", "plt.plot(x, y, 'g-')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "我们把这个过程改为迭代 10 次：" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "a = 0.04433333333333332, b = 0.03033333333333333, loss = 52.07898433333334, diff = 2.566015666666658\n", "a = 0.08762444444444442, b = 0.059919999999999994, loss = 49.63396493034815, diff = 2.4450194029851886\n", "a = 0.12989817037037035, b = 0.08877788888888888, loss = 47.30423372457149, diff = 2.3297312057766604\n", "a = 0.17117875530864196, b = 0.11692446162962962, loss = 45.08435194574915, diff = 2.219881778822341\n", "a = 0.21148986477053494, b = 0.14437676279358025, loss = 42.969137414855254, diff = 2.115214530893894\n", "a = 0.25085459939642357, b = 0.17115142985127244, loss = 40.95365243850431, diff = 2.0154849763509475\n", "a = 0.2892955084413038, b = 0.1972647028921288, loss = 39.03319227448755, diff = 1.9204601640167596\n", "a = 0.32683460293836075, b = 0.22273243411212781, loss = 37.20327414148798, diff = 1.8299181329995662\n", "a = 0.3634933685482668, b = 0.24757009707495625, loss = 35.45962674730114, diff = 1.7436473941868442\n", "a = 0.3992927781017128, b = 0.27179279575205734, loss = 33.79818031109894, diff = 1.6614464362021977\n" ] } ], "source": [ "for i in range(10):\n", " a_new = update_a(X, Y, a, b, eta)\n", " b_new = update_b(X, Y, a, b, eta)\n", " loss_new = loss(X, Y, a_new, b_new)\n", " diff = abs(loss(X, Y, a, b) - loss(X, Y, a_new, b_new))\n", " print(\"a = {0}, b = {1}, loss = {2}, diff = {3}\".format(a_new, b_new, loss_new, diff))\n", " a = a_new\n", " b = b_new" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "可以看到 loss 和 diff 都变得越来越小，说明损失函数在不断收敛，画出现在的图像：" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.plot(X, Y, 'k.')\n", "\n", "x = np.linspace(0, 10, 10)\n", "y = a_new*x+b_new\n", "\n", "plt.plot(x, y, 'g-')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "比迭代一次靠谱多了，我们再继续这个过程，迭代 1000 次：" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "scrolled": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "a = 2.083663961231524, b = 0.9559983763104101, loss = 0.010980851812161636, diff = 5.500906925493335e-06\n" ] } ], "source": [ "for i in range(1000):\n", " a_new = update_a(X, Y, a, b, eta)\n", " b_new = update_b(X, Y, a, b, eta)\n", " loss_new = loss(X, Y, a_new, b_new)\n", " diff = abs(loss(X, Y, a, b) - loss(X, Y, a_new, b_new))\n", " a = a_new\n", " b = b_new\n", " \n", "print(\"a = {0}, b = {1}, loss = {2}, diff = {3}\".format(a_new, b_new, loss_new, diff))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "diff 的值已经趋近于 0，说明已经达到收敛，这个时候的图形，就是我们根据梯度下降法求得的结果：" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.plot(X, Y, 'k.')\n", "\n", "x = np.linspace(0, 10, 10)\n", "y = a_new*x+b_new\n", "\n", "plt.plot(x, y, 'g-')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 多元线性回归的梯度下降\n", "\n", "上面是对一元线性回归的梯度下降法的介绍，回到线性回归的一般形式：\n", "\n", "$$\n", "\\hat{y} = \\theta_0 + \\theta_1x_1 + \\theta_2x_2 + ... + \\theta_nx_n = \\theta^Tx\n", "$$\n", "\n", "其对应的损失函数为：\n", "\n", "$$\n", "loss = J(\\theta_0, \\theta_1, \\dots, \\theta_n) = J(\\theta) = \\frac{1}{2m} \\sum_{i=1}^{m}(y^{(i)} - \\theta^Tx^{(i)})^2\n", "$$\n", "\n", "对 $J(\\theta)$ 进行求导，得到参数 $\\theta$ 的更新公式：\n", "\n", "$$\n", "\\left\\{\n", "\\begin{align}\n", "\\theta_0 &:= \\theta_0 - \\eta \\frac{1}{m} \\sum_{i=1}^{m}(y^{(i)} - \\theta^Tx^{(i)})x_0^{(i)} \\\\\n", "\\theta_1 &:= \\theta_1 - \\eta \\frac{1}{m} \\sum_{i=1}^{m}(y^{(i)} - \\theta^Tx^{(i)})x_1^{(i)} \\\\\n", "\\theta_2 &:= \\theta_2 - \\eta \\frac{1}{m} \\sum_{i=1}^{m}(y^{(i)} - \\theta^Tx^{(i)})x_2^{(i)} \\\\\n", "\\vdots \\\\\n", "\\theta_n &:= \\theta_n - \\eta \\frac{1}{m} \\sum_{i=1}^{m}(y^{(i)} - \\theta^Tx^{(i)})x_n^{(i)} \\\\\n", "\\end{align}\n", "\\right.\n", "$$\n", "\n", "其中，$m$ 表示样本个数，$n$ 表示特征维度，$x^{(i)}$ 表示第 i 个样本，$x^{(i)}_n$ 表示第 i 个样本的第 n 个特征。\n", "为了方便表达和计算，损失函数也可以写成下面的矩阵形式：\n", "\n", "$$\n", "loss = J(\\theta) = \\frac{1}{2m} (\\bf{y}-\\rm{X}\\theta)^T (\\bf{y}-\\rm{X}\\theta)\n", "$$\n", "\n", "根据前面的矩阵求导，得到梯度下降的更新公式为：\n", "\n", "$$\n", "\\begin{aligned}\n", "\\theta &:= \\theta - \\eta \\frac{\\partial}{\\partial\\theta}J(\\theta) \\\\\n", " &= \\theta - \\eta \\frac{1}{m} X^T(\\rm{X}\\theta-\\bf{y})\n", "\\end{aligned}\n", "$$\n", "\n", "TODO：使用矩阵的迹（Trace）求矩阵导数。\n", "\n", "我们把上面的代码改成矩阵计算的形式：" ] }, { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "theta: 0.9333333336913473 2.099999999741958\n", "loss: [[0.01]]\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "# 计算损失函数 J 的梯度\n", "def gradient_function(theta, X, y):\n", " diff = np.dot(X, theta) - y\n", " return (1./y.size) * np.dot(np.transpose(X), diff)\n", "\n", "# 梯度下降\n", "def gradient_descent(X, y, eta, theta_init, max_iter):\n", " theta = theta_init\n", " i = 0\n", " gradient = gradient_function(theta, X, y)\n", " while not np.all(np.absolute(gradient) <= 1e-10):\n", " theta = theta - eta * gradient\n", "# print('theta:', theta[0,0], theta[1,0])\n", " i += 1\n", " if i > max_iter:\n", " break\n", " gradient = gradient_function(theta, X, y)\n", "# print('gradient:', gradient[0,0], gradient[1,0])\n", " return theta\n", "\n", "# 损失函数\n", "def loss_function(theta, X, y):\n", " diff = np.dot(X, theta) - y\n", " return (1./2*y.size) * np.dot(np.transpose(diff), diff)\n", "\n", "# 数据集\n", "X0 = np.ones((3, 1))\n", "X1 = np.array([0,1,2]).reshape(-1, 1)\n", "X = np.hstack((X0, X1))\n", "y = np.array([0.9, 3.1, 5.1]).reshape(-1, 1)\n", "\n", "# 学习率\n", "eta = 0.01\n", "\n", "# 最大迭代次数\n", "max_iter = 10000\n", "\n", "# 初始值\n", "theta_init = np.array([0, 0]).reshape(-1, 1)\n", "\n", "# 使用梯度下降求解线性回归\n", "theta = gradient_descent(X, y, eta, theta_init, max_iter)\n", "print('theta:', theta[0,0], theta[1,0])\n", "print('loss:', loss_function(theta, X, y))\n", "\n", "# 绘制图形\n", "plt.plot(X1, y, 'k.')\n", "x = np.linspace(0, 10, 10)\n", "y = theta[0,0]*x+theta[1,0]\n", "plt.plot(x, y, 'g-')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 疑惑一\n", "\n", "为什么上面两种方法求出来的解不一样？" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 疑惑二\n", "\n", "在前面介绍正规方程法解一元线性回归时，使用了下面的数据集，这个数据集不知道为什么用梯度下降法无法收敛？\n", "\n", "```\n", "X = np.array([39.93, 42.05, 43.18, 44.68, 49.87, 53.57])\n", "Y = np.array([199, 290, 298, 310, 399, 420])\n", "```" ] }, { "cell_type": "code", "execution_count": 63, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "theta: 0.7489393245009788 7.078701795630663\n", "loss: [[32892.8042478]]\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "# 计算损失函数 J 的梯度\n", "def gradient_function(theta, X, y):\n", " diff = np.dot(X, theta) - y\n", " return (1./y.size) * np.dot(np.transpose(X), diff)\n", "\n", "# 梯度下降\n", "def gradient_descent(X, y, eta, theta_init, max_iter):\n", " theta = theta_init\n", " i = 0\n", " gradient = gradient_function(theta, X, y)\n", " while not np.all(np.absolute(gradient) <= 1e-10):\n", " theta = theta - eta * gradient\n", " i += 1\n", " if i > max_iter:\n", " break\n", " gradient = gradient_function(theta, X, y)\n", " return theta\n", "\n", "# 损失函数\n", "def loss_function(theta, X, y):\n", " diff = np.dot(X, theta) - y\n", " return (1./2*y.size) * np.dot(np.transpose(diff), diff)\n", "\n", "# 数据集\n", "X0 = np.ones((6, 1))\n", "X1 = np.array([39.93, 42.05, 43.18, 44.68, 49.87, 53.57]).reshape(-1, 1)\n", "X = np.hstack((X0, X1))\n", "y = np.array([199, 290, 298, 310, 399, 420]).reshape(-1, 1)\n", "\n", "# 学习率\n", "eta = 0.00001\n", "\n", "# 最大迭代次数\n", "max_iter = 10000\n", "\n", "# 初始值\n", "theta_init = np.array([1, 1]).reshape(-1, 1)\n", "\n", "# 使用梯度下降求解线性回归\n", "theta = gradient_descent(X, y, eta, theta_init, max_iter)\n", "print('theta:', theta[0,0], theta[1,0])\n", "print('loss:', loss_function(theta, X, y))\n", "\n", "# 绘制图形\n", "plt.plot(X1, y, 'k.')\n", "x = np.linspace(30, 60, 100)\n", "y = theta[0,0]*x+theta[1,0]\n", "plt.plot(x, y, 'g-')\n", "plt.show()" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.1" } }, "nbformat": 4, "nbformat_minor": 2 }