{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.4.1\n"
     ]
    }
   ],
   "source": [
    "import torch\n",
    "\n",
    "print(torch.__version__)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2.3 自动求梯度\n",
    "## 2.3.1 概念\n",
    "上一节介绍的`Tensor`是这个包的核心类，如果将其属性`.requires_grad`设置为`True`，它将开始追踪(track)在其上的所有操作。完成计算后，可以调用`.backward()`来完成所有梯度计算。此`Tensor`的梯度将累积到`.grad`属性中。\n",
    "> 注意在调用`.backward()`时，如果`Tensor`是标量，则不需要为`backward()`指定任何参数；否则，需要指定一个求导变量。\n",
    "\n",
    "如果不想要被继续追踪，可以调用`.detach()`将其从追踪记录中分离出来，这样就可以防止将来的计算被追踪。此外，还可以用`with torch.no_grad()`将不想被追踪的操作代码块包裹起来，这种方法在评估模型的时候很常用，因为在评估模型时，我们并不需要计算可训练参数（`requires_grad=True`）的梯度。\n",
    "\n",
    "`Function`是另外一个很重要的类。`Tensor`和`Function`互相结合就可以构建一个记录有整个计算过程的非循环图。每个`Tensor`都有一个`.grad_fn`属性，该属性即创建该`Tensor`的`Function`（除非用户创建的`Tensor`s时设置了`grad_fn=None`）。\n",
    "\n",
    "下面通过一些例子来理解这些概念。\n",
    "\n",
    "## 2.3.2 `Tensor`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([[1., 1.],\n",
      "        [1., 1.]], requires_grad=True)\n",
      "None\n"
     ]
    }
   ],
   "source": [
    "x = torch.ones(2, 2, requires_grad=True)\n",
    "print(x)\n",
    "print(x.grad_fn)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([[3., 3.],\n",
      "        [3., 3.]], grad_fn=<AddBackward>)\n",
      "<AddBackward object at 0x10ed634a8>\n"
     ]
    }
   ],
   "source": [
    "y = x + 2\n",
    "print(y)\n",
    "print(y.grad_fn)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "注意x是直接创建的，所以它没有`grad_fn`, 而y是通过一个加法操作创建的，所以它有一个为`<AddBackward>`的`grad_fn`。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True False\n"
     ]
    }
   ],
   "source": [
    "print(x.is_leaf, y.is_leaf)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([[27., 27.],\n",
      "        [27., 27.]], grad_fn=<MulBackward>) tensor(27., grad_fn=<MeanBackward1>)\n"
     ]
    }
   ],
   "source": [
    "z = y * y * 3\n",
    "out = z.mean()\n",
    "print(z, out)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "通过`.requires_grad_()`来用in-place的方式改变`requires_grad`属性："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "False\n",
      "True\n",
      "<SumBackward0 object at 0x10ed63c50>\n"
     ]
    }
   ],
   "source": [
    "a = torch.randn(2, 2) # 缺失情况下默认 requires_grad = False\n",
    "a = ((a * 3) / (a - 1))\n",
    "print(a.requires_grad) # False\n",
    "a.requires_grad_(True)\n",
    "print(a.requires_grad) # True\n",
    "b = (a * a).sum()\n",
    "print(b.grad_fn)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2.3.3 梯度 \n",
    "\n",
    "因为`out`是一个标量，所以调用`backward()`时不需要指定求导变量："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([[4.5000, 4.5000],\n",
      "        [4.5000, 4.5000]])\n"
     ]
    }
   ],
   "source": [
    "out.backward() # 等价于 out.backward(torch.tensor(1.))\n",
    "print(x.grad)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们令`out`为 $o$ , 因为\n",
    "$$\n",
    "o=\\frac14\\sum_{i=1}^4z_i=\\frac14\\sum_{i=1}^43(x_i+2)^2\n",
    "$$\n",
    "所以\n",
    "$$\n",
    "\\frac{\\partial{o}}{\\partial{x_i}}\\bigr\\rvert_{x_i=1}=\\frac{9}{2}=4.5\n",
    "$$\n",
    "所以上面的输出是正确的。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "数学上，如果有一个函数值和自变量都为向量的函数 $\\vec{y}=f(\\vec{x})$, 那么 $\\vec{y}$ 关于 $\\vec{x}$ 的梯度就是一个雅可比矩阵（Jacobian matrix）:\n",
    "\n",
    "$$\n",
    "J=\\left(\\begin{array}{ccc}\n",
    "   \\frac{\\partial y_{1}}{\\partial x_{1}} & \\cdots & \\frac{\\partial y_{1}}{\\partial x_{n}}\\\\\n",
    "   \\vdots & \\ddots & \\vdots\\\\\n",
    "   \\frac{\\partial y_{m}}{\\partial x_{1}} & \\cdots & \\frac{\\partial y_{m}}{\\partial x_{n}}\n",
    "   \\end{array}\\right)\n",
    "$$\n",
    "\n",
    "而``torch.autograd``这个包就是用来计算一些雅克比矩阵的乘积的。例如，如果 $v$ 是一个标量函数的 $l=g\\left(\\vec{y}\\right)$ 的梯度：\n",
    "\n",
    "$$\n",
    "v=\\left(\\begin{array}{ccc}\\frac{\\partial l}{\\partial y_{1}} & \\cdots & \\frac{\\partial l}{\\partial y_{m}}\\end{array}\\right)\n",
    "$$\n",
    "\n",
    "那么根据链式法则我们有 $l$ 关于 $\\vec{x}$ 的雅克比矩阵就为:\n",
    "\n",
    "$$\n",
    "v \\cdot J=\\left(\\begin{array}{ccc}\\frac{\\partial l}{\\partial y_{1}} & \\cdots & \\frac{\\partial l}{\\partial y_{m}}\\end{array}\\right) \\left(\\begin{array}{ccc}\n",
    "   \\frac{\\partial y_{1}}{\\partial x_{1}} & \\cdots & \\frac{\\partial y_{1}}{\\partial x_{n}}\\\\\n",
    "   \\vdots & \\ddots & \\vdots\\\\\n",
    "   \\frac{\\partial y_{m}}{\\partial x_{1}} & \\cdots & \\frac{\\partial y_{m}}{\\partial x_{n}}\n",
    "   \\end{array}\\right)=\\left(\\begin{array}{ccc}\\frac{\\partial l}{\\partial x_{1}} & \\cdots & \\frac{\\partial l}{\\partial x_{n}}\\end{array}\\right)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "注意：grad在反向传播过程中是累加的(accumulated)，这意味着每一次运行反向传播，梯度都会累加之前的梯度，所以一般在反向传播之前需把梯度清零。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([[5.5000, 5.5000],\n",
      "        [5.5000, 5.5000]])\n",
      "tensor([[1., 1.],\n",
      "        [1., 1.]])\n"
     ]
    }
   ],
   "source": [
    "# 再来反向传播一次，注意grad是累加的\n",
    "out2 = x.sum()\n",
    "out2.backward()\n",
    "print(x.grad)\n",
    "\n",
    "out3 = x.sum()\n",
    "x.grad.data.zero_()\n",
    "out3.backward()\n",
    "print(x.grad)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([[2., 4.],\n",
      "        [6., 8.]], grad_fn=<ViewBackward>)\n"
     ]
    }
   ],
   "source": [
    "x = torch.tensor([1.0, 2.0, 3.0, 4.0], requires_grad=True)\n",
    "y = 2 * x\n",
    "z = y.view(2, 2)\n",
    "print(z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "现在 `y` 不是一个标量，所以在调用`backward`时需要传入一个和`y`同形的权重向量进行加权求和得到一个标量。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([2.0000, 0.2000, 0.0200, 0.0020])\n"
     ]
    }
   ],
   "source": [
    "v = torch.tensor([[1.0, 0.1], [0.01, 0.001]], dtype=torch.float)\n",
    "z.backward(v)\n",
    "\n",
    "print(x.grad)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "再来看看中断梯度追踪的例子："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor(1., requires_grad=True) True\n",
      "tensor(1., grad_fn=<PowBackward0>) True\n",
      "tensor(1.) False\n",
      "tensor(2., grad_fn=<ThAddBackward>) True\n"
     ]
    }
   ],
   "source": [
    "x = torch.tensor(1.0, requires_grad=True)\n",
    "y1 = x ** 2 \n",
    "with torch.no_grad():\n",
    "    y2 = x ** 3\n",
    "y3 = y1 + y2\n",
    "    \n",
    "print(x, x.requires_grad)\n",
    "print(y1, y1.requires_grad)\n",
    "print(y2, y2.requires_grad)\n",
    "print(y3, y3.requires_grad)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor(2.)\n"
     ]
    }
   ],
   "source": [
    "y3.backward()\n",
    "print(x.grad)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "为什么是2呢？$ y_3 = y_1 + y_2 = x^2 + x^3$，当 $x=1$ 时 $\\frac {dy_3} {dx}$ 不应该是5吗？事实上，由于 $y_2$ 的定义是被`torch.no_grad():`包裹的，所以与 $y_2$ 有关的梯度是不会回传的，只有与 $y_1$ 有关的梯度才会回传，即 $x^2$ 对 $x$ 的梯度。\n",
    "\n",
    "上面提到，`y2.requires_grad=False`，所以不能调用 `y2.backward()`。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# y2.backward() # 会报错 RuntimeError: element 0 of tensors does not require grad and does not have a grad_fn"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "如果我们想要修改`tensor`的数值，但是又不希望被`autograd`记录（即不会影响反向传播），那么我么可以对`tensor.data`进行操作."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([1.])\n",
      "False\n",
      "tensor([100.], requires_grad=True)\n",
      "tensor([2.])\n"
     ]
    }
   ],
   "source": [
    "x = torch.ones(1,requires_grad=True)\n",
    "\n",
    "print(x.data) # 还是一个tensor\n",
    "print(x.data.requires_grad) # 但是已经是独立于计算图之外\n",
    "\n",
    "y = 2 * x\n",
    "x.data *= 100 # 只改变了值，不会记录在计算图，所以不会影响梯度传播\n",
    "\n",
    "y.backward()\n",
    "print(x) # 更改data的值也会影响tensor的值\n",
    "print(x.grad)"
   ]
  }
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