chapter3_NN/linear-regression-gradient-descend.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 线性模型和梯度下降\n",
    "这是神经网络的第一课，我们会学习一个非常简单的模型，线性回归，同时也会学习一个优化算法-梯度下降法，对这个模型进行优化。线性回归是监督学习里面一个非常简单的模型，同时梯度下降也是深度学习中应用最广的优化算法，我们将从这里开始我们的深度学习之旅"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 一元线性回归\n",
    "一元线性模型非常简单，假设我们有变量 $x_i$ 和目标 $y_i$，每个 i 对应于一个数据点，希望建立一个模型\n",
    "\n",
    "$$\n",
    "\\hat{y}_i = w x_i + b\n",
    "$$\n",
    "\n",
    "$\\hat{y}_i$ 是我们预测的结果，希望通过 $\\hat{y}_i$ 来拟合目标 $y_i$，通俗来讲就是找到这个函数拟合 $y_i$ 使得误差最小，即最小化\n",
    "\n",
    "$$\n",
    "\\frac{1}{n} \\sum_{i=1}^n(\\hat{y}_i - y_i)^2\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "那么如何最小化这个误差呢？\n",
    "\n",
    "这里需要用到**梯度下降**，这是我们接触到的第一个优化算法，非常简单，但是却非常强大，在深度学习中被大量使用，所以让我们从简单的例子出发了解梯度下降法的原理"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 梯度下降法\n",
    "在梯度下降法中，我们首先要明确梯度的概念，随后我们再了解如何使用梯度进行下降。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 梯度\n",
    "梯度在数学上就是导数，如果是一个多元函数，那么梯度就是偏导数。比如一个函数f(x, y)，那么 f 的梯度就是 \n",
    "\n",
    "$$\n",
    "(\\frac{\\partial f}{\\partial x},\\ \\frac{\\partial f}{\\partial y})\n",
    "$$\n",
    "\n",
    "可以称为 grad f(x, y) 或者 $\\nabla f(x, y)$。具体某一点 $(x_0,\\ y_0)$ 的梯度就是 $\\nabla f(x_0,\\ y_0)$。\n",
    "\n",
    "下面这个图片是 $f(x) = x^2$ 这个函数在 x=1 处的梯度\n",
    "\n",
    "![](https://ws3.sinaimg.cn/large/006tNc79ly1fmarbuh2j3j30ba0b80sy.jpg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "梯度有什么意义呢？从几何意义来讲，一个点的梯度值是这个函数变化最快的地方，具体来说，对于函数 f(x, y)，在点 $(x_0, y_0)$ 处，沿着梯度 $\\nabla f(x_0,\\ y_0)$ 的方向，函数增加最快，也就是说沿着梯度的方向，我们能够更快地找到函数的极大值点，或者反过来沿着梯度的反方向，我们能够更快地找到函数的最小值点。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 梯度下降法\n",
    "有了对梯度的理解，我们就能了解梯度下降发的原理了。上面我们需要最小化这个误差，也就是需要找到这个误差的最小值点，那么沿着梯度的反方向我们就能够找到这个最小值点。\n",
    "\n",
    "我们可以来看一个直观的解释。比如我们在一座大山上的某处位置，由于我们不知道怎么下山，于是决定走一步算一步，也就是在每走到一个位置的时候，求解当前位置的梯度，沿着梯度的负方向，也就是当前最陡峭的位置向下走一步，然后继续求解当前位置梯度，向这一步所在位置沿着最陡峭最易下山的位置走一步。这样一步步的走下去，一直走到觉得我们已经到了山脚。当然这样走下去，有可能我们不能走到山脚，而是到了某一个局部的山峰低处。\n",
    "\n",
    "类比我们的问题，就是沿着梯度的反方向，我们不断改变 w 和 b 的值，最终找到一组最好的 w 和 b 使得误差最小。\n",
    "\n",
    "在更新的时候，我们需要决定每次更新的幅度，比如在下山的例子中，我们需要每次往下走的那一步的长度，这个长度称为学习率，用 $\\eta$ 表示，这个学习率非常重要，不同的学习率都会导致不同的结果，学习率太小会导致下降非常缓慢，学习率太大又会导致跳动非常明显，可以看看下面的例子\n",
    "\n",
    "![](https://ws2.sinaimg.cn/large/006tNc79ly1fmgn23lnzjg30980gogso.gif)\n",
    "\n",
    "可以看到上面的学习率较为合适，而下面的学习率太大，就会导致不断跳动\n",
    "\n",
    "最后我们的更新公式就是\n",
    "\n",
    "$$\n",
    "w := w - \\eta \\frac{\\partial f(w,\\ b)}{\\partial w} \\\\\n",
    "b := b - \\eta \\frac{\\partial f(w,\\ b)}{\\partial b}\n",
    "$$\n",
    "\n",
    "通过不断地迭代更新，最终我们能够找到一组最优的 w 和 b，这就是梯度下降法的原理。\n",
    "\n",
    "最后可以通过这张图形象地说明一下这个方法\n",
    "\n",
    "![](https://ws3.sinaimg.cn/large/006tNc79ly1fmarxsltfqj30gx091gn4.jpg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "上面是原理部分，下面通过一个例子来进一步学习线性模型"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<torch._C.Generator at 0x112959630>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import torch\n",
    "import numpy as np\n",
    "from torch.autograd import Variable\n",
    "\n",
    "torch.manual_seed(2017)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# 读入数据 x 和 y\n",
    "x_train = np.array([[3.3], [4.4], [5.5], [6.71], [6.93], [4.168],\n",
    "                    [9.779], [6.182], [7.59], [2.167], [7.042],\n",
    "                    [10.791], [5.313], [7.997], [3.1]], dtype=np.float32)\n",
    "\n",
    "y_train = np.array([[1.7], [2.76], [2.09], [3.19], [1.694], [1.573],\n",
    "                    [3.366], [2.596], [2.53], [1.221], [2.827],\n",
    "                    [3.465], [1.65], [2.904], [1.3]], dtype=np.float32)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x11565e588>]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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/kPTTvuF7JD2V339K0h9saVFjUMlwX812U9JuSfvLraQy/lbSn0s6X3YhFfMbklYk/X1+\nyOpbtj9VdlFVEBHvSfobScuSTko6HRH/Wm5VlfOZiDiZ3/+JpM+UWcwoVDrcbV8u6TlJD0fEmbLr\nKZvtuyWdiogDZddSQZdIulHSExGxW9J/K4E/rUchP358j7L/AH9d0qds/1G5VVVXZC2EE99GWNlw\nt71dWbB3I+L5suupiFsl/b7t45L+UdLttv+h3JIq44SkExFx4S+8Z5WFPaTflfQfEbESEf8r6XlJ\nt5RcU9X8l+2rJCnfniq5nk2rZLjbtrJjp8ci4rGy66mKiPjLiNgVEU1lH4h9PyJYgUmKiJ9Ietf2\nZ/OhL0g6WmJJVbIs6Wbbjfx36wviw+Z+L0r6cn7/y5L+qcRaRqKS4a5shXq/spXpofx2V9lFofL+\nRFLX9mFJN0j665LrqYT8r5lnJR2U9Kay3/vkzsgsyvYzkv5N0mdtn7D9x5K+IekO2z9W9pfON8qs\ncRQ4QxUAElTVlTsAYBMIdwBIEOEOAAki3AEgQYQ7ACSIcAeABBHuAJAgwh0AEvR/pcuwDbjnoq8A\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1146f2438>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 画出图像\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "plt.plot(x_train, y_train, 'bo')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# 转换成 Tensor\n",
    "x_train = torch.from_numpy(x_train)\n",
    "y_train = torch.from_numpy(y_train)\n",
    "\n",
    "# 定义参数 w 和 b\n",
    "w = Variable(torch.randn(1), requires_grad=True) # 随机初始化\n",
    "b = Variable(torch.zeros(1), requires_grad=True) # 使用 0 进行初始化"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# 构建线性回归模型\n",
    "x_train = Variable(x_train)\n",
    "y_train = Variable(y_train)\n",
    "\n",
    "def linear_model(x):\n",
    "    return x * w + b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y_ = linear_model(x_train)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "经过上面的步骤我们就定义好了模型，在进行参数更新之前，我们可以先看看模型的输出结果长什么样"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x11577da58>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1155f3be0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x_train.data.numpy(), y_train.data.numpy(), 'bo', label='real')\n",
    "plt.plot(x_train.data.numpy(), y_.data.numpy(), 'ro', label='estimated')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**思考：红色的点表示预测值，似乎排列成一条直线，请思考一下这些点是否在一条直线上？**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "这个时候需要计算我们的误差函数，也就是\n",
    "\n",
    "$$\n",
    "\\frac{1}{n} \\sum_{i=1}^n(\\hat{y}_i - y_i)^2\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# 计算误差\n",
    "def get_loss(y_, y):\n",
    "    return torch.mean((y_ - y_train) ** 2)\n",
    "\n",
    "loss = get_loss(y_, y_train)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Variable containing:\n",
      " 153.3520\n",
      "[torch.FloatTensor of size 1]\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# 打印一下看看 loss 的大小\n",
    "print(loss)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "定义好了误差函数，接下来我们需要计算 w 和 b 的梯度了，这时得益于 PyTorch 的自动求导，我们不需要手动去算梯度，有兴趣的同学可以手动计算一下，w 和 b 的梯度分别是\n",
    "\n",
    "$$\n",
    "\\frac{\\partial}{\\partial w} = \\frac{2}{n} \\sum_{i=1}^n x_i(w x_i + b - y_i) \\\\\n",
    "\\frac{\\partial}{\\partial b} = \\frac{2}{n} \\sum_{i=1}^n (w x_i + b - y_i)\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# 自动求导\n",
    "loss.backward()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Variable containing:\n",
      " 161.0043\n",
      "[torch.FloatTensor of size 1]\n",
      "\n",
      "Variable containing:\n",
      " 22.8730\n",
      "[torch.FloatTensor of size 1]\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# 查看 w 和 b 的梯度\n",
    "print(w.grad)\n",
    "print(b.grad)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# 更新一次参数\n",
    "w.data = w.data - 1e-2 * w.grad.data\n",
    "b.data = b.data - 1e-2 * b.grad.data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "更新完成参数之后，我们再一次看看模型输出的结果"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x11588b358>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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0s9YCj+l094y7ZxobG8tdJ4ACWPFZm0qaVeLuf5K0WdLSeMoBEFXa9qVG5USZVdJoZm/J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\nAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1156dc6d8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y_ = linear_model(x_train)\n",
    "plt.plot(x_train.data.numpy(), y_train.data.numpy(), 'bo', label='real')\n",
    "plt.plot(x_train.data.numpy(), y_.data.numpy(), 'ro', label='estimated')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "从上面的例子可以看到，更新之后红色的线跑到了蓝色的线下面，没有特别好的拟合蓝色的真实值，所以我们需要在进行几次更新"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epoch: 0, loss: 3.1357719898223877\n",
      "epoch: 1, loss: 0.3550889194011688\n",
      "epoch: 2, loss: 0.30295443534851074\n",
      "epoch: 3, loss: 0.30131956934928894\n",
      "epoch: 4, loss: 0.3006229102611542\n",
      "epoch: 5, loss: 0.29994693398475647\n",
      "epoch: 6, loss: 0.299274742603302\n",
      "epoch: 7, loss: 0.2986060082912445\n",
      "epoch: 8, loss: 0.2979407012462616\n",
      "epoch: 9, loss: 0.29727882146835327\n"
     ]
    }
   ],
   "source": [
    "for e in range(10): # 进行 10 次更新\n",
    "    y_ = linear_model(x_train)\n",
    "    loss = get_loss(y_, y_train)\n",
    "    \n",
    "    w.grad.zero_() # 记得归零梯度\n",
    "    b.grad.zero_() # 记得归零梯度\n",
    "    loss.backward()\n",
    "    \n",
    "    w.data = w.data - 1e-2 * w.grad.data # 更新 w\n",
    "    b.data = b.data - 1e-2 * b.grad.data # 更新 b \n",
    "    print('epoch: {}, loss: {}'.format(e, loss.data[0]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x11598e0f0>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1158466d8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y_ = linear_model(x_train)\n",
    "plt.plot(x_train.data.numpy(), y_train.data.numpy(), 'bo', label='real')\n",
    "plt.plot(x_train.data.numpy(), y_.data.numpy(), 'ro', label='estimated')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "经过 10 次更新，我们发现红色的预测结果已经比较好的拟合了蓝色的真实值。\n",
    "\n",
    "现在你已经学会了你的第一个机器学习模型了，再接再厉，完成下面的小练习。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**小练习：**\n",
    "\n",
    "重启 notebook 运行上面的线性回归模型，但是改变训练次数以及不同的学习率进行尝试得到不同的结果"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 多项式回归模型"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "下面我们更进一步，讲一讲多项式回归。什么是多项式回归呢？非常简单，根据上面的线性回归模型\n",
    "\n",
    "$$\n",
    "\\hat{y} = w x + b\n",
    "$$\n",
    "\n",
    "这里是关于 x 的一个一次多项式，这个模型比较简单，没有办法拟合比较复杂的模型，所以我们可以使用更高次的模型，比如\n",
    "\n",
    "$$\n",
    "\\hat{y} = w_0 + w_1 x + w_2 x^2 + w_3 x^3 + \\cdots\n",
    "$$\n",
    "\n",
    "这样就能够拟合更加复杂的模型，这就是多项式模型，这里使用了 x 的更高次，同理还有多元回归模型，形式也是一样的，只是出了使用 x，还是更多的变量，比如 y、z 等等，同时他们的 loss 函数和简单的线性回归模型是一致的。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "首先我们可以先定义一个需要拟合的目标函数，这个函数是个三次的多项式"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y = 0.90 + 0.50 * x + 3.00 * x^2 + 2.40 * x^3\n"
     ]
    }
   ],
   "source": [
    "# 定义一个多变量函数\n",
    "\n",
    "w_target = np.array([0.5, 3, 2.4]) # 定义参数\n",
    "b_target = np.array([0.9]) # 定义参数\n",
    "\n",
    "f_des = 'y = {:.2f} + {:.2f} * x + {:.2f} * x^2 + {:.2f} * x^3'.format(\n",
    "    b_target[0], w_target[0], w_target[1], w_target[2]) # 打印出函数的式子\n",
    "\n",
    "print(f_des)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们可以先画出这个多项式的图像"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x1158f5e48>"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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8vtK/Y9AliYh8jgI/SrWhML9+axV//MenDO7ahj9cOUxfzIpIXFLgR2FveQ23\nTJvPh5/u5IpR3Zl8wQCdLSsicUuBX0/rd+7nW0/NY+Oucn71zZMZn98t6JJERI5IgV8P89btYtLT\nBTjw7A2jGNWrfdAliYgcVVRTSMzsV2a2wswWm9lLZpZ90HN3m1mRma00s3OjLzU+vLywmCsf/Zi2\nmc146TunKexFJGFEO2dwFjDI3U8GVgF3A5jZAOAyYCBwHvAHM0vowW1353dvr+aOGYsY1iObF79z\nKnkdWgZdlojIMYsq8N39LXevjTycA+RG7l8ETHf3KndfCxQBI6NpK0jhsHP/a4X89u1VXDqsK09f\nP4rsTK1wKSKJJZZnBV0PvBm53xXYeNBzmyLb/oWZTTKzAjMrKCkpiWE5sVEbCnPnC4t56sN13HB6\nHr8ZP0QnU4lIQjrql7Zm9jbQ6RBP3evur0T2uReoBaYdbwHuPgWYApCfn+/H+/qGVFUb4o7pi3hz\n6VbuGNuH28/uo8sOikjCOmrgu/vYIz1vZtcBFwBnu/uBwC4GDp6nmBvZljAqqkPc/Ox83ltVwv+7\nYAA3nJ4XdEkiIlGJdpbOecCdwNfdvfygp14FLjOz5maWB/QB5kbTVmMqr67l2ifn8v7qEn75jZMV\n9iLSJEQ7D/+/gObArMhQxxx3/7a7LzOz54FC6oZ6bnX3UJRtNYrKmhA3Ti2gYN0ufnfZUC4c0iXo\nkkREYiKqwHf33kd47kHgwWjev7FV1Ya4+Zn5fLRmJ7+dcIrCXkSaFE03iagJhbl12kL+saqEhy4d\nzMVDDzmpSEQkYSnwqZt6ecf0Rby9fBu/uGggE0d0D7okEZGYS/rAD4edO2cu5n+XbOGnXzuJq8f0\nDLokEZEGkfSB/9BfV/DiwmJ+OK4vN36pV9DliIg0mKQO/Cc/WMuU99ZwzZge3PaVw37/LCLSJCRt\n4L+5ZAs/f72QcwZ05L4LB+oMWhFp8pIy8AvW7eL2GYsY2i2b318+lNQUhb2INH1JF/hF2/dxw9QC\ncrNb8Ni1I3RJQhFJGkkV+Dv3VXHdk3NJTzWe+tZI2rXUEscikjyS5hKHNaEwt0xbQElZFc/fPIbu\n7TODLklEpFElTeD//LVC5q7dxe8uO4Uh3bKP/gIRkSYmKYZ0/vzxBp6Zs56bz+zFRadoyQQRSU5N\nPvDnrt3F5FeWcmbfHO48t3/Q5YiIBKZJB37xngpueXY+3dplavqliCS9Jhv4lTUhbn6mgKraMI9e\nM5w2LdJdmmnEAAAFUElEQVSDLklEJFBN9kvb+18rZGlxKY9dk0/vE7KCLkdEJHBNsof/yqJinpu7\ngZvP7MXYAR2DLkdEJC40ucBfU7KPe15cwvAebfnROf2CLkdEJG40qcCvrAlx658Xkp6Wwn9ePpT0\n1CZ1eCIiUWlSY/i/eL2Q5VtKeeK6fLpktwi6HBGRuNJkusCvfbKZaR9v4OYzevGV/hq3FxH5opgE\nvpn90MzczDoctO1uMysys5Vmdm4s2jmctTv2c/eLSxjWPZsfnatxexGRQ4l6SMfMugHnABsO2jYA\nuAwYCHQB3jazvu4eira9Q0lLMYZ2z+ahb5yscXsRkcOIRTr+FrgT8IO2XQRMd/cqd18LFAEjY9DW\nIXVrl8kzN4yiq8btRUQOK6rAN7OLgGJ3/+QLT3UFNh70eFNk26HeY5KZFZhZQUlJSTTliIjIERx1\nSMfM3gY6HeKpe4F7qBvOqTd3nwJMAcjPz/ej7C4iIvV01MB397GH2m5mg4E84JPIBcBzgQVmNhIo\nBrodtHtuZJuIiASk3kM67r7E3U9w957u3pO6YZth7r4VeBW4zMyam1ke0AeYG5OKRUSkXhrkxCt3\nX2ZmzwOFQC1wa0PN0BERkWMTs8CP9PIPfvwg8GCs3l9ERKKjSesiIklCgS8ikiTMPX5mQppZCbA+\nirfoAOyIUTlBairHATqWeNRUjgN0LAf0cPeco+0UV4EfLTMrcPf8oOuIVlM5DtCxxKOmchygYzle\nGtIREUkSCnwRkSTR1AJ/StAFxEhTOQ7QscSjpnIcoGM5Lk1qDF9ERA6vqfXwRUTkMJpU4JvZL8xs\nsZktMrO3zKxL0DXVl5n9ysxWRI7nJTPLDrqm+jKz8Wa2zMzCZpZwMyrM7LzIlduKzOyuoOupLzN7\nwsy2m9nSoGuJlpl1M7N3zKww8m/r9qBrqg8zyzCzuWb2SeQ47m/Q9prSkI6ZtXb30sj97wED3P3b\nAZdVL2Z2DvB3d681s4cB3P0nAZdVL2Z2EhAG/gT8yN0LAi7pmJlZKrAKGEfdAoHzgMvdvTDQwurB\nzM4A9gFPu/ugoOuJhpl1Bjq7+wIzywLmAxcn2t+L1S013NLd95lZOvA+cLu7z2mI9ppUD/9A2Ee0\n5PNX4Uoo7v6Wu9dGHs6hbonphOTuy919ZdB11NNIoMjd17h7NTCduiu6JRx3fw/YFXQdseDuW9x9\nQeR+GbCcw1xkKZ55nX2Rh+mRPw2WW00q8AHM7EEz2whcCUwOup4YuR54M+giktQxX71NgmFmPYGh\nwMfBVlI/ZpZqZouA7cAsd2+w40i4wDezt81s6SH+XATg7ve6ezdgGnBbsNUe2dGOJbLPvdQtMT0t\nuEqP7liORSTWzKwV8AJwxxc+4ScMdw+5+ynUfYofaWYNNtzWIOvhN6TDXYHrEKYBbwD3NWA5UTna\nsZjZdcAFwNke51+2HMffS6LR1dviVGTM+wVgmru/GHQ90XL3PWb2DnAe0CBfrCdcD/9IzKzPQQ8v\nAlYEVUu0zOw84E7g6+5eHnQ9SWwe0MfM8sysGXAZdVd0kwBFvux8HFju7o8EXU99mVnOgRl4ZtaC\nuskBDZZbTW2WzgtAP+pmhKwHvu3uCdkbM7MioDmwM7JpTgLPOLoE+E8gB9gDLHL3c4Ot6tiZ2fnA\nfwCpwBORi/skHDN7DjiLulUZtwH3ufvjgRZVT2Z2OvBPYAl1/98B7nH3N4Kr6viZ2cnAVOr+baUA\nz7v7zxusvaYU+CIicnhNakhHREQOT4EvIpIkFPgiIklCgS8ikiQU+CIiSUKBLyKSJBT4IiJJQoEv\nIpIk/j8hjlpJApitKgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1158f5dd8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 画出这个函数的曲线\n",
    "x_sample = np.arange(-3, 3.1, 0.1)\n",
    "y_sample = b_target[0] + w_target[0] * x_sample + w_target[1] * x_sample ** 2 + w_target[2] * x_sample ** 3\n",
    "\n",
    "plt.plot(x_sample, y_sample, label='real curve')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "接着我们可以构建数据集，需要 x 和 y，同时是一个三次多项式，所以我们取了 $x,\\ x^2, x^3$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# 构建数据 x 和 y\n",
    "# x 是一个如下矩阵 [x, x^2, x^3]\n",
    "# y 是函数的结果 [y]\n",
    "\n",
    "x_train = np.stack([x_sample ** i for i in range(1, 4)], axis=1)\n",
    "x_train = torch.from_numpy(x_train).float() # 转换成 float tensor\n",
    "\n",
    "y_train = torch.from_numpy(y_sample).float().unsqueeze(1) # 转化成 float tensor "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "接着我们可以定义需要优化的参数，就是前面这个函数里面的 $w_i$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# 定义参数和模型\n",
    "w = Variable(torch.randn(3, 1), requires_grad=True)\n",
    "b = Variable(torch.zeros(1), requires_grad=True)\n",
    "\n",
    "# 将 x 和 y 转换成 Variable\n",
    "x_train = Variable(x_train)\n",
    "y_train = Variable(y_train)\n",
    "\n",
    "def multi_linear(x):\n",
    "    return torch.mm(x, w) + b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们可以画出没有更新之前的模型和真实的模型之间的对比"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x115b8cc50>"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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oTfmC7eSTI+SKcWwJdcKvDWwsdD8d+EvhF4hIX6AvwGkRVXM18mVna1/xypW6\nhOe332q/cXr6oa+rXl2H+J1xhp6J16mjLdSTT9bx3QW3Rb7Glp2tB/nll4Pbpk3aJVB4y8g4/r7i\n4rTZXLasfpPEx0O5+INN+FKlDm4iB5vshbfcXG2u5+Toz9nZumXu19tAxMcf/KYrV+7Qn8uV07gL\nbgs+Q+FTkvjDPkvhz+TcoVt+/sHPUbBlZ+tpVcGpSMG2e7cm79279dTkaGrWhIQEOPNMXfM1IUH/\nIZx1lj524omB/X5MifL8oq1zbjwwHiAxMdEd5+XGT87pYIhFi3RbskSX8CzoWilXThtml16qgyXO\nPlu3s87ys1Wan69FrNasOXiqkJqqt5s2/bmz/qSTNLnUrKmn/zVr6rdItWpQpYrOpKxSRU8dKlXS\nfo+KFTUhhrIl6Zwmzf37ddu3T/tssrL+3Fd1vK3gPYVv//jj4L6zsg72FRXuNyroAy+quLhDvzQq\nVDh0q1hRu1cKTsUKfq+Fv71PPlm/6aNsEe9YF+qEvwmoU+h+gu8xE2LOaX6dO1dnpC9apGfooI2y\nv/wF/v1v7Udu2lSTvN8Tc7KydCTGN9/oacLKlXrRrnDLsWpVOOccvRJ5xhnaYX3aabolJITvEDyR\ng61vrxS04PPy9Iu04Fbk0K1UqYP9Y8YcQagT/tdAfRGphyb6HsD1IT5mzPrjD13zYe5c3QpGv9Wt\nCx07wvnn69agQQBT6wufKnz1lW4rVx48VahaVb9BbrlFC1k1aKCnCiedFIyPGJsKJ3RjAhDShO+c\nyxWR/sBH6LDMCc65VaE8ZqzZvh2Sk7XUyMcfay9A5craNTNkiC5iceaZAR7k5591BmTBVtDJX6mS\ndr/8+9+QlKSjMhIS7GKdMWEq5H34zrkPgA9CfZxY8scfMH26lrn97DM9u69bV6vDXnut5t6A6qbs\n3687nj0b3n//4EzIGjX0wt0ll2gFroBOFYwxJc3zi7amaPLytHE9caK25vft056SIUPg73+HZs0C\nbFjv3AnvvQf/9396qrB3r17JbdcOBgzQU4YGDaxbwZgIZgk/zG3dqgtKv/yyDoeuXBl699bqsK1b\nB5jkd+/WJD91qib53Fwdc/mPf2h1wksusaJVxkQRS/hhavlyGDcOJk/Wfvn27eHJJ6Fz5wBXJsrJ\n0W6aiRO1OmF2to6YufdeLT9rNcaNiVqW8MOIc9qjMno0fPmlDpfu00f75s89N8Cdr12rC5i+/rqe\nNtSqBf2t4s/WAAAOQUlEQVT66YoYAZ8qGGMigSX8MJCfr2s9PPaYjnCsWxfGjtWV26pUCWDHOTm6\nWtBzz+k3SOnS0KmTDpns2DFCVsQwxgSL/Y/3UF6e5uPHHtNZr2efDZMmwfXXB5iLt2+H8ePh+ed1\nVutZZ8GYMXDjjTqD1RgTkyzhe2TuXLjvPp2Q2qABvP22dqEHNMpx3TpN7G++qbNf27fXK75XXGGj\na4wxlvBL2vffaznwDz+EevVgyhRdczSgfLxqFYwYoaNt4uN1lM1dd2m9cWOM8bGEX0I2b4YHHoAJ\nE3Ro5dixcMcdAZZo+eYbGD5cB+ZXqqSnDPfea902xpgjsoQfYvn52qsyeLBOlhowAIYN0yKQflu3\nDoYO1QsAlSvrN8mAAVavxhhzTJbwQ+jbb+G227QUcbt28MILWpXSb5s3w6OP6gXZ8uXh4Yd1PdDK\nlYMVsjEmilnCD4G9e+GRR7Tbplo1vYZ6/fUBDHXPyNCLsU89pXVubr9dW/XWdWOMKQZL+EG2aJFe\nM12/Hm69FUaNCqD7xjmtknbvvTq8smtXvTgb0GmCMSZW2Vi9IMnO1r75Cy/U8fULFmj9G7+T/Zo1\ncNllOhP25JN14tS0aZbsjTF+s4QfBGvWQJs22vju3Vtny150kZ87y8iAgQN18ZDly3WW7Ndf68ol\nxhgTAOvSCYBzOpn1vvt0VOS772o9er/Nm6f9QBs2aPmDJ57QGvTGGBME1sL3U0aGXojt31+rCH/3\nXQDJfvdu6NtXZ8aWLQtffAGvvGLJ3hgTVJbw/bBmjRaYnDZNG+GzZ8Mpp/i5sw8/hEaNtJLl/ffD\nihVwwQVBjdcYYyDAhC8iY0RkrYh8KyIzRaRKoeeGiEiaiPwgIh0CDzU8TJmiy7hu365rhgwe7GdZ\nhL17tVV/xRVw4ok6vGf0aB1fb4wxIRBoC/9joJFzrgmwDhgCICINgB5AQ6Aj8LyIRPTip7m5cM89\n0LMnNG2qVQ0uvdTPna1YAYmJ2m0zaJBenP3LX4IarzHGHC6ghO+cm+ucy/XdXQIk+H7uAkxxzu13\nzv0EpAGtAzmWl/bsgS5d4JlntCbZggVQu7YfO8rPh6ef1uS+e7depB05MsCCOsYYUzTBHKVzMzDV\n93Nt9AugQLrvsT8Rkb5AX4DTTjstiOEExy+/6Johq1drTZy+ff3c0ZYtuhDthx/qOoWvvgrVqwcz\nVGOMOabjJnwRmQcc6ZLkUOfcLN9rhgK5wFvFDcA5Nx4YD5CYmOiK+/5Q+vprzc2Zmbr8a/v2fu5o\n8WK47jrYsUPH1f/rX7akoDGmxB034TvnLjvW8yJyE9AJaOecK0jYm4A6hV6W4HssYsycCb16abma\n+fN1kZJicw5efFErWdapA199pRcAjDHGA4GO0ukIDAQ6O+cyCz2VDPQQkbIiUg+oDywN5Fgl6eWX\n4e9/19z81Vd+Jvt9++Dmm7Xo/WWXQUqKJXtjjKcCHaXzP+AE4GMRWSEiLwI451YB04DVwIdAP+dc\nXoDHKhFjxhwcLTl/vpaxKbYNG+Cvf4WJE+HBB3WgftWqwQ7VGGOKJaCLts65s47x3AhgRCD7L0nO\n6ZoiTzwBPXroYuLx8X7saMkS7fjPzobkZLj66qDHaowx/rCZtuhoyX79NNnfdpvWr/cr2U+bpnUW\nTjxR+4Is2RtjwkjMJ/zcXK1f/8ILOgfqhRcgrrhTxJyDxx/XUsYtW2or/5xzQhKvMcb4K6arZebl\n6dD4t97SfD1kiB87yc7WFahee02rqU2YYBOpjDFhKWZb+Pn5Wok4oGS/Zw9cdZUm+4ce0r4gS/bG\nmDAVky38/HxtlE+cqOuA+5Xst2+HK6/UOjgTJ+rKJ8YYE8ZiLuE7B3feqWPt//MfHTVZbOnpcPnl\n8OOPOkPLLs4aYyJATCV853Q98Oef19Lzw4f7UeFg3TqtsfD77/DRRwGsZWiMMSUrphL+iBFa8XLA\nABg1yo9kv2IFdOig3xwLFkCLFqEI0xhjQiJmLtpOmAAPPAA33ghjx/qR7FNSdIx92bKwcKEle2NM\nxImJFv7s2VouoUMHrUpc7BWqUlK0G6dKFW3Zn356KMI0xpiQivoW/pIl0K0bNG8OM2ZAmTLF3IEl\ne2NMlIjqhL92rQ6TP/VUeP99qFSpmDuwZG+MiSJRm/C3bIGOHaF0aR1MU+yql5bsjTFRJir78Pfv\nh2uvha1b4fPP4cwzi7mD777TcfaW7I0xUSTqEr5zeoF28WKYPh0SE4u5gx9/1GRfvjx88okle2NM\n1Ii6hP/UU/D661oy4brrivnm337TbpzsbD01qFcvFCEaY4wnoirhf/ABDBwIXbvqmPti2blTW/Zb\ntmjLvmHDkMRojDFeiZqEv3q1rlTVrJnWMivWWPu9e3U4z7p1+q3RunWowjTGGM8EZZSOiPxbRJyI\nVC/02BARSRORH0SkQzCOczQ7dmj9sgoVYNYsvS2ynBz4299g6VKYPBnatQtZnMYY46WAW/giUge4\nHPil0GMNgB5AQ+BUYJ6InB2qhcw3bNBu9/fegzp1ivFG57RO8ty58MormviNMSZKBaOF/zQwEHCF\nHusCTHHO7XfO/QSkASHrJ2nRAtLSICmpmG98/PGDRXZuuSUksRljTLgIKOGLSBdgk3Nu5WFP1QY2\nFrqf7nvsSPvoKyIpIpKybds2v2Mp9kJTkyfDsGHQqxc88ojfxzXGmEhx3C4dEZkHnHKEp4YC/0G7\nc/zmnBsPjAdITEx0x3l5cCxcqIvZtm2r1dSKXTrTGGMiz3ETvnPusiM9LiKNgXrAStGEmQAsF5HW\nwCagcG96gu8x761bB9dco2PsZ860NWiNMTHD7y4d59x3zrmTnXN1nXN10W6bFs65zUAy0ENEyopI\nPaA+sDQoEQdi505dhzYuTodfVqvmdUTGGFNiQjIO3zm3SkSmAauBXKBfqEboFFluLnTvDhs3an2c\nM87wNBxjjClpQUv4vlZ+4fsjgBHB2n/ABg2CefO0z75NG6+jMcaYEhe15ZEP8cYbuq5h//5w881e\nR2OMMZ6I/oSfkgJ9+sDFF2vSN8aYGBXdCX/LFi2Mf8opMG2aH+sbGmNM9Iia4ml/kpOjZTN37IAv\nv4QaNbyOyBhjPBW9CX/wYJ1g9fbbuoK5McbEuOjs0pk1S/vr+/WDnj29jsYYY8JC9CX8n36C3r2h\nZUtd/soYYwwQbQl//37o1k1/nj7dyiYYY0wh0dWHf999Ogxz5kxbj9YYYw4TPS386dPhf/+De+7R\n4mjGGGMOER0JPzVVFzBJSoKRI72OxhhjwlJ0JPzSpTXZT50K8fFeR2OMMWEpOvrw69XTdWmNMcYc\nVXS08I0xxhyXJXxjjIkRlvCNMSZGWMI3xpgYYQnfGGNiRMAJX0TuFJG1IrJKREYXenyIiKSJyA8i\n0iHQ4xhjjAlMQMMyReQSoAvQ1Dm3X0RO9j3eAOgBNAROBeaJyNmeL2RujDExLNAW/r+Akc65/QDO\nua2+x7sAU5xz+51zPwFpQOsAj2WMMSYAgU68Ohu4UERGAFnAfc65r4HawJJCr0v3PfYnItIX6Ou7\nmyEiPwQQT3VgewDvDxfR8jnAPks4ipbPAfZZCpxelBcdN+GLyDzglCM8NdT3/mpAEtAKmCYiZxQj\nSJxz44HxxXnP0YhIinMuMRj78lK0fA6wzxKOouVzgH2W4jpuwnfOXXa050TkX8C7zjkHLBWRfPRb\nahNQp9BLE3yPGWOM8UigffjvAZcAiMjZQDx6SpIM9BCRsiJSD6gPLA3wWMYYYwIQaB/+BGCCiHwP\nZAO9fa39VSIyDVgN5AL9SmiETlC6hsJAtHwOsM8SjqLlc4B9lmIRzc/GGGOinc20NcaYGBFVCV9E\nHhORb0VkhYjMFZFTvY7JXyIyxjeD+VsRmSkiVbyOyV8i0tU3EztfRCJuRIWIdPTNGE8TkcFex+Mv\nEZkgIlt9XbARTUTqiMinIrLa929rgNcx+UNEyonIUhFZ6fscj4T0eNHUpSMiJzrn/vD9fBfQwDl3\nu8dh+UVELgc+cc7lisgoAOfcII/D8ouInAfkAy+hczVSPA6pyEQkDlgHtEfnk3wN9HTOrfY0MD+I\nSFsgA3jdOdfI63gCISK1gFrOueUicgKwDLgm0v4uIiJARedchoiUAb4ABjjnlhznrX6JqhZ+QbL3\nqQhE7LeZc26ucy7Xd3cJOrQ1Ijnn1jjnAplQ56XWQJpz7kfnXDYwBZ1JHnGcc58DO72OIxicc785\n55b7ft4DrOEokzvDmVMZvrtlfFvI8lZUJXwAERkhIhuBXsCDXscTJDcDc7wOIkbVBjYWun/UWePG\nGyJSF2gOfOVtJP4RkTgRWQFsBT52zoXsc0RcwheReSLy/RG2LgDOuaHOuTrAW0B/b6M9tuN9Ft9r\nhqJDW9/yLtLjK8pnMSbYRKQS8A5w92Fn+BHDOZfnnGuGnsW3FpGQdbdF3CLmx5r5e5i3gA+Ah0IY\nTkCO91lE5CagE9DOhfnFlmL8XSKNzRoPU74+73eAt5xz73odT6Ccc7tE5FOgIxCSC+sR18I/FhGp\nX+huF2CtV7EESkQ6AgOBzs65TK/jiWFfA/VFpJ6IxKNlv5M9jinm+S52vgqscc6N9Toef4lIjYIR\neCJSHh0cELK8FW2jdN4BzkFHhGwAbnfORWRrTETSgLLADt9DSyJ4xNG1wLNADWAXsMI5FzGL4ojI\nlcAzQBwwwTk3wuOQ/CIik4GL0XpXW4CHnHOvehqUn0Tkr8BC4Dv0/zvAf5xzH3gXVfGJSBNgEvpv\nqxQwzTn3aMiOF00J3xhjzNFFVZeOMcaYo7OEb4wxMcISvjHGxAhL+MYYEyMs4RtjTIywhG+MMTHC\nEr4xxsQIS/jGGBMj/h9fRUynBRhSVgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x115846320>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 画出更新之前的模型\n",
    "y_pred = multi_linear(x_train)\n",
    "\n",
    "plt.plot(x_train.data.numpy()[:, 0], y_pred.data.numpy(), label='fitting curve', color='r')\n",
    "plt.plot(x_train.data.numpy()[:, 0], y_sample, label='real curve', color='b')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "可以发现，这两条曲线之间存在差异，我们计算一下他们之间的误差"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Variable containing:\n",
      " 413.9843\n",
      "[torch.FloatTensor of size 1]\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# 计算误差，这里的误差和一元的线性模型的误差是相同的，前面已经定义过了 get_loss\n",
    "loss = get_loss(y_pred, y_train)\n",
    "print(loss)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# 自动求导\n",
    "loss.backward()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Variable containing:\n",
      " -34.1391\n",
      "-146.6133\n",
      "-215.9148\n",
      "[torch.FloatTensor of size 3x1]\n",
      "\n",
      "Variable containing:\n",
      "-27.0838\n",
      "[torch.FloatTensor of size 1]\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# 查看一下 w 和 b 的梯度\n",
    "print(w.grad)\n",
    "print(b.grad)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# 更新一下参数\n",
    "w.data = w.data - 0.001 * w.grad.data\n",
    "b.data = b.data - 0.001 * b.grad.data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x1164c6d30>"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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g1NTf/3xcnBbBv/lmaNhQk3yjRvrCTMQIdcKvBmzP9n0icEn2A0SkD9AHoEaN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\n/JAtQgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x115bb9438>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 画出更新一次之后的模型\n",
    "y_pred = multi_linear(x_train)\n",
    "\n",
    "plt.plot(x_train.data.numpy()[:, 0], y_pred.data.numpy(), label='fitting curve', color='r')\n",
    "plt.plot(x_train.data.numpy()[:, 0], y_sample, label='real curve', color='b')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "因为只更新了一次，所以两条曲线之间的差异仍然存在，我们进行 100 次迭代"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epoch 20, Loss: 73.67840\n",
      "epoch 40, Loss: 17.97097\n",
      "epoch 60, Loss: 4.94101\n",
      "epoch 80, Loss: 1.87171\n",
      "epoch 100, Loss: 1.12812\n"
     ]
    }
   ],
   "source": [
    "# 进行 100 次参数更新\n",
    "for e in range(100):\n",
    "    y_pred = multi_linear(x_train)\n",
    "    loss = get_loss(y_pred, y_train)\n",
    "    \n",
    "    w.grad.data.zero_()\n",
    "    b.grad.data.zero_()\n",
    "    loss.backward()\n",
    "    \n",
    "    # 更新参数\n",
    "    w.data = w.data - 0.001 * w.grad.data\n",
    "    b.data = b.data - 0.001 * b.grad.data\n",
    "    if (e + 1) % 20 == 0:\n",
    "        print('epoch {}, Loss: {:.5f}'.format(e+1, loss.data[0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "可以看到更新完成之后 loss 已经非常小了，我们画出更新之后的曲线对比"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x1164e8278>"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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EA68DrYF44AcRmaOqcdnZrskijwd27XKO/G3a5HQ3N292UmnvXueUj9ObEsTh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\neQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x115ad6128>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 画出更新之后的结果\n",
    "y_pred = multi_linear(x_train)\n",
    "\n",
    "plt.plot(x_train.data.numpy()[:, 0], y_pred.data.numpy(), label='fitting curve', color='r')\n",
    "plt.plot(x_train.data.numpy()[:, 0], y_sample, label='real curve', color='b')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "可以看到，经过 100 次更新之后，可以看到拟合的线和真实的线已经完全重合了"
   ]
  },
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    "**小练习：上面的例子是一个三次的多项式，尝试使用二次的多项式去拟合它，看看最后能做到多好**\n",
    "\n",
    "**提示：参数 `w = torch.randn(2, 1)`，同时重新构建 x 数据集**"
   ]
  }
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