Lesson5基本优化思想与最小二乘法.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lesson5.基本优化思想与最小二乘法"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import torch"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 一、简单线性回归的机器学习建模思路\n",
    "\n",
    "### 1.回顾简单线性回归建模问题"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[1., 2.],\n",
       "        [3., 4.]])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = torch.arange(1, 5).reshape(2, 2).float()\n",
    "A\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x208407af608>]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 绘制点图查看两个点的位置\n",
    "plt.plot(A[:,0], A[:, 1], 'o')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "如果更进一步，我们希望在二维空间中找到一条直线，来拟合这两个点，也就是所谓的构建一个线性回归模型，我们可以设置线性回归方程如下：\n",
    "$$y=ax+b$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.转化为优化问题\n",
    "\n",
    "可以利用逆矩阵求解，但不通用。\n",
    "\n",
    "最优化问题的转化分为两步，\n",
    "- 其一是确定优化数值指标\n",
    "- 其二则是确定优化目标函数\n",
    "- 两者相辅相成\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "就是希望方程的预测值和真实值相差较小。假设真实的y值用y表示，预测值用ŷ表示，带入a、b参数，则有数值表示如下：\n",
    "\n",
    "|$ x^{(i)} $| $ y^{(i)} $ | $ ŷ^{(i)} $ |\n",
    "| :------:| :------: | :------ |\n",
    "| 1 | 2 | a+b |\n",
    "| 3 | 4 | 3a+b |\n",
    "| ŷ表示 | 对应预测值 | \n",
    "\n",
    "$$ŷ_1 = 1*a + b  = a + b$$\n",
    "$$ŷ_2 = 3*a + b  = 3a + b$$\n",
    "\n",
    "希望真实值和预测值尽可以接近，使用误差平方和SSE\n",
    "- SSE代表着真实值和预测值之间的差值（误差平方和），因此也被称为损失函数（\n",
    "  - 只有拟合系数的模型，才有损失函数"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.最优化问题的求解方法\n",
    "\n",
    "- 图形展示目标函数\n",
    "\n",
    "&emsp;&emsp;为了更好的讨论目标函数（SSE）求最小值的过程，对于上述二元函数来说，我们可以将其展示在三维空间内。此处我们可以使用Python中matplotlib包和Axes3D函数进行三维图像绘制"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "from matplotlib import pyplot as plt\n",
    "from mpl_toolkits.mplot3d import Axes3D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.arange(-1,3,0.05)\n",
    "y = np.arange(-1,3,0.05)\n",
    "a,b = np.meshgrid(x,y)\n",
    "\n",
    "SSE = (2-a-b)**2 +(4-3*a-b)**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#————————————————————————————绘制函数图像————————————————————————————\n",
    "ax = plt.axes(projection='3d')\n",
    "\n",
    "ax.plot_surface(a, b, SSE, cmap='rainbow')\n",
    "#  绘制等高线\n",
    "ax.contour(a, b, SSE, zdir='z', offset=0, cmap=\"rainbow\")  #生成z方向投影，投到x-y平面\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- 函数的凹凸性\n",
    "\n",
    "不难看出，目标函数是个整体看起来“向下凸”的函数，函数的凹凸性是函数的重要性质，首先我们给出凸函数的一般定义，对于任意一个函数，如果函数f(x)上存在任意两个点，$x_1, x_2$，且      \n",
    "$$ (f(x_1) + f(x_2))/2 >= f((x_1 + x_2)/2) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- **凸函数，存在全局的最小值**\n",
    "\n",
    "    凸函数求解最小值的一般方法：      \n",
    "1. 对于一元函数，如果存在导数为0的点，则该点就是最小值点；      \n",
    "2. 对于多元函数，如果存在某一点，使得函数的各个自变量的偏导数都为0，则该点就是最小值点。 \n",
    "        \n",
    "因此，对于凸函数的最小值求解，最基本的出发点就是寻找导数为0的点。而最小二乘法也是基于偏导函数取值为0联立的方程组进行的求解。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4.机器学习建模一般流程\n",
    "\n",
    "**优化建模的一般思路：**\n",
    "- **Step 1**：提出基本模型  \n",
    "- **Step 2**：确定损失函数和目标函数\n",
    "  - 损失函数不是模型，是模型参数所组成的函数\n",
    "- **Step 3**：根据目标函数特性，选择优化方法，求解目标函数"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 二、最小二乘法\n",
    "\n",
    "### 1.最小二乘法的代数表示方法\n",
    "\n",
    "首先，假设多元线性方程有如下形式\n",
    "$$ f(x) = w_1x_1+w_2x_2+...+w_dx_d+b $$\n",
    "令$w = (w_1,w_2,...w_d)$，$x = (x_1,x_2,...x_d)$，则上式可写为\n",
    "$$ f(x) = w^Tx+b $$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> 我们将线性回归自变量系数命名为w，其实是weight的简写，意为自变量的权重。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "优化目标可以写为\n",
    "$$ SSE = \\sum^m_{i=1}(f(x_i)-y_i)^2 = E_(w,b) $$\n",
    "通过偏导为0求得最终结果的最小二乘法求解过程为：\n",
    "$$ \\begin{align} \\frac{\\partial{SSE_(w,b)}}{\\partial{(w)}} \n",
    "& = 2(w\\sum^m_{i=1}x^2_i - \\sum^m_{i=1}(y_i-b)x_i) = 0  \\end{align} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\begin{align} \\frac{\\partial{SSE_(w,b)}}{\\partial{(b)}} \n",
    "& = 2(mb - \\sum^m_{i=1}(y_i-wx_i)) = 0  \\end{align}  $$ \n",
    "\n",
    "从而\n",
    "$$ w = \\frac{\\sum^m_{i=1}y_i(x_i-\\bar{x})\n",
    "}{\\sum^m_{i=1}x^2_i-\\frac{1}{m}(\\sum^m_{i=1}x_i)^2\n",
    "} $$\n",
    "\n",
    "$$ b = \\frac{1}{m}\\sum^m_{i=1}(y_i-wx_i)\n",
    "$$\n",
    "\n",
    "其中，$\\bar x = \\frac{1}{m}\\sum^m_{i=1}x_i，x_i$为x的均值，并且$(x_i,y_i)$代表二维空间中的点。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.最小二乘法的矩阵表示形式\n",
    "\n",
    "原方程组可表示为$ A * X^T = B $\n",
    "\n",
    "更为一般的情况下，多元线性回归方程为\n",
    "$$ f(x) = w_1x_1+w_2x_2+...+w_dx_d+b $$\n",
    "令\n",
    "$$\\hat w = (w_1,w_2,...,w_d,b)$$\n",
    "$$\\hat x = (x_1,x_2,...,x_d,1)$$\n",
    "- $\\hat w$：方程系数所组成的向量，并且我们将自变量系数和截距放到了一个向量中，此处$\\hat w$就相当于前例中的a、b组成的向量(a,b)；\n",
    "- $\\hat x$：方程自变量和1共同组成的向量；\n",
    "因此，方程可表示为\n",
    "  $$f(x) = \\hat w * \\hat x^T$$\n",
    "\n",
    "\n",
    "另外，我们将所有自变量的值放在一个矩阵中，并且和此前A矩阵类似，为了捕捉截距，添加一列全为1的列在矩阵的末尾，设总共有m组取值，则\n",
    "$$ X = \n",
    "\\left [\\begin{array}{cccc}\n",
    "x_{11} &x_{12} &... &x_{1d} &1 \\\\\n",
    "x_{21} &x_{22} &... &x_{2d} &1 \\\\\n",
    "... &... &... &... &1 \\\\\n",
    "x_{m1} &x_{m2} &... &x_{md} &1 \\\\\n",
    "\\end{array}\\right]\n",
    "$$\n",
    "\n",
    "对应到前例中的A矩阵，A矩阵就是拥有一个自变量、两个取值的X矩阵。令y为因变量的取值，则有\n",
    "\n",
    "$$ y =  \\left [\\begin{array}{cccc}\n",
    "y_1 \\\\\n",
    "y_2 \\\\\n",
    ". \\\\\n",
    ". \\\\\n",
    ". \\\\\n",
    "y_m \\\\\n",
    "\\end{array}\\right] $$\n",
    "\n",
    "\n",
    "此时，SSE可表示为：\n",
    "$$SSE = ||y - X\\hat w^T||_2^2 = (y - X\\hat w^T)^T(y - X\\hat w^T) = E(\\hat w)$$\n",
    "\n",
    "根据最小二乘法的求解过程，令$E(\\hat w)$对$\\hat w$求导方程取值为0，有\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\frac{E(\\hat w)}{\\partial{\\boldsymbol{\\hat w}}} \n",
    "&= \\frac{\\partial{||\\boldsymbol{y} - \\boldsymbol{X\\hat w^T}||_2}^2}{\\partial{\\boldsymbol{\\hat w}}}\n",
    "\\\\\n",
    "&= \\frac{\\partial(\\boldsymbol{y} - \\boldsymbol{X\\hat w^T})^T(\\boldsymbol{y} - \\boldsymbol{X\\hat w^T})}{\\partial{\\boldsymbol{\\hat w}}}\n",
    "\\\\\n",
    "\\\\ \\because &\\ (A-B)^T = A^T - B^T并且(AB)^T = B^T*A^T \\\\ \\\\\n",
    "\\therefore& 上式=\\frac{\\partial(\\boldsymbol{y}^T - \\boldsymbol{\\hat w X^T})(\\boldsymbol{y} - \\boldsymbol{X\\hat w^T})}{\\partial{\\boldsymbol{\\hat w}}}\n",
    "\\\\\n",
    "&=\\frac{\\partial(\\boldsymbol{y}^T\\boldsymbol{y} - \\boldsymbol{\\hat w X^Ty}-\\boldsymbol{y}^T\\boldsymbol{X \\hat w^T} +\\boldsymbol{\\hat wX^T}\\boldsymbol{X\\hat w^T})}\n",
    "{\\partial{\\boldsymbol{\\hat w}}}\\\\ \n",
    "&= 0 - \\boldsymbol{X^Ty} - \\boldsymbol{X^Ty} + 2\\boldsymbol{X^TX\\hat w ^T}\\\\\n",
    "&= \\boldsymbol{X^TX\\hat w^T} - \\boldsymbol{X^Ty} = 0\n",
    "\\\\\n",
    "&值得注意的是，在矩阵求导中，a为常数，有如下规则：\n",
    "\\begin{equation}\n",
    "\\\\\\frac{\\partial{a}}{\\partial{A}} = 0, \\ \\ \\ \\frac{\\partial{A^TB^TC}}{\\partial{A}} = B^TC, \\ \\ \\ \\frac{\\partial{C^TBA}}{\\partial{A}} = B^TC, \\ \\ \\ \n",
    "\\frac{\\partial{A^TBA}}{\\partial{A}} = (B+B^T)A\n",
    "\\end{equation}\n",
    "\\end{aligned}\n",
    "$$ \n",
    "\n",
    "进一步可得\n",
    "$$X^TX\\hat w^T = X^Ty$$\n",
    "\n",
    "要使得此式有解，等价于$X^TX$（也被称为矩阵的交叉乘积crossprod存在逆矩阵，若存在，则可解出\n",
    "\n",
    "$$\\hat w ^T = (X^TX)^{-1}X^Ty$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.最小二乘法的简单实现"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[1., 1.],\n",
       "        [3., 1.]])"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X= torch.tensor([[1,1],[3,1]]).float()\n",
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[2.],\n",
       "        [4.]])"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = torch.tensor([2,4]).float().reshape(2,1)\n",
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[1., 3.],\n",
       "        [2., 4.]])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.t()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[1.0000],\n",
       "        [1.0000]])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 计算权重 (X^T*)^-1*X^T*y\n",
    "w = torch.mm(torch.mm(torch.inverse(torch.mm(X.t(),X)),X.t()),y)\n",
    "w"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- 调用最小二乘法函数`torch.lstsq(y, X)`进行求解\n",
    "  - 先输入y 在输入X\n",
    "  - 第一个参数是因变量张量\n",
    "  - 第二个参数是自变量张量\n",
    "  - 同时返回结果还包括QR矩阵分解的结果\n",
    "\n",
    "- `linalg.norm(对象，p)`求函数或举证的Lp范数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "g:\\py\\lib\\site-packages\\ipykernel_launcher.py:1: UserWarning: torch.lstsq is deprecated in favor of torch.linalg.lstsq and will be removed in a future PyTorch release.\n",
      "torch.linalg.lstsq has reversed arguments and does not return the QR decomposition in the returned tuple (although it returns other information about the problem).\n",
      "To get the qr decomposition consider using torch.linalg.qr.\n",
      "The returned solution in torch.lstsq stored the residuals of the solution in the last m - n columns of the returned value whenever m > n. In torch.linalg.lstsq, the residuals in the field 'residuals' of the returned named tuple.\n",
      "The unpacking of the solution, as in\n",
      "X, _ = torch.lstsq(B, A).solution[:A.size(1)]\n",
      "should be replaced with\n",
      "X = torch.linalg.lstsq(A, B).solution (Triggered internally at  C:\\actions-runner\\_work\\pytorch\\pytorch\\builder\\windows\\pytorch\\aten\\src\\ATen\\native\\BatchLinearAlgebra.cpp:4169.)\n",
      "  \"\"\"Entry point for launching an IPython kernel.\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "torch.return_types.lstsq(\n",
       "solution=tensor([[1.0000],\n",
       "        [1.0000]]),\n",
       "QR=tensor([[-3.1623, -1.2649],\n",
       "        [ 0.7208, -0.6325]]))"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "torch.lstsq(y,X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- `linalg.norm`函数求向量或矩阵的范数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([-1.,  2.])"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = torch.tensor([-1,2.])\n",
    "t"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(2.2361)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "torch.sqrt(torch.tensor(5.))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(3.)"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 输入参数，求解L1范数，个元素的绝对值之和\n",
    "torch.linalg.norm(t, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4.反向验证导数为零\n",
    "\n",
    "- 目的是为了引出autograd模块\n",
    "\n",
    "&emsp;&emsp;当然，我们也可以反向验证，看下损失函数SSE在a=1，b=1时偏导数是否都为0。此时就需要借助PyTorch中的autograd模块来进行偏导计算。\n",
    "\n",
    "- autograd模块是PyTorch中的自动微分模块\n",
    "- 需要在创建Tensor时，通过设置requires_grad属性为True、规定张量可微分即可。\n",
    "- 查看可微分性`.requires_grad`\n",
    "  - 默认为False,需要可微性需要在建立张量时设置 requires_grad=True\n",
    "- `torch.autograd.grad(函数，[参数1,参数2])`计算偏导\n",
    "  - 一个参数就求导数\n",
    "  - 多个参数时求偏导"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(1., requires_grad=True)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = torch.tensor(1.,requires_grad = True)  # 默认是False\n",
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "此时a就是一个可微分的张量，requires_grad是a的一个属性，可以查看可以修改。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 查看可微分性\n",
    "a.requires_grad"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 修改可微分性\n",
    "a.requires_grad = False\n",
    "a.requires_grad"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "a.requires_grad = True"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(1., requires_grad=True)"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = torch.tensor(1.,requires_grad = True)\n",
    "b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "创建损失函数\n",
    "$$(y_1 - ŷ_1)^2 + (y_2 - ŷ_2)^2 $$     \n",
    "$$=  (2 - a - b)^2 + (4 - 3a - b)^2$$    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "sse = torch.pow((2 - a - b), 2) + torch.pow((4 - 3 * a - b), 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "使用`torch.autograd.grad`进行偏导运算，并输出a=1、b=1时偏导数的值"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(tensor(-0.), tensor(-0.))"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "torch.autograd.grad(sse,[a, b])\n",
    "\n",
    "# 至此，也可验证(1,1)是损失函数的最小值点。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- `torch.autograd.grad(对象，[参数1，参数2,...])`函数      \n",
    "torch.autograd.grad是通用微分函数，当只输入一个自变量时计算结果就是导数，输入多个自变量时则会计算偏导数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(1., requires_grad=True)"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = torch.tensor(1.,requires_grad=True)\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = x ** 2 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(tensor(2.),)"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 导数计算结果\n",
    "torch.autograd.grad(y,x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "当然，微分计算作为神经网络优化的核心，autograd模块的功能远不仅于此，更多的微分运算，我们将在下一节继续讲解。"
   ]
  }
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