07 自动求导.ipynb

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"
    }
   },
   "cell_type": "markdown",
   "id": "97227e8c-405c-4ba6-9134-5e7f155cd73c",
   "metadata": {},
   "source": [
    "![image.png](attachment:a3a5af63-5e11-46cc-aa40-ba1899f4c2c8.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "c3ba63fb-cfbe-48b7-af1a-389c5a3cb7e4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([0., 1., 2., 3.])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import torch\n",
    "\n",
    "x = torch.arange(4.0)\n",
    "\n",
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7dbd5c60-cb1c-488d-a704-a60b41f1c19c",
   "metadata": {},
   "source": [
    "在计算y 关于x的梯度之前，需要一个地方存储梯度"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "e3d09fb4-ae24-44da-8494-9d4eb140bc10",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(tensor([0., 1., 2., 3.], requires_grad=True), None)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.requires_grad_(True) # 等价 x = torch.arange(4.0, requires_grad=True)\n",
    "\n",
    "x, x.grad"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "38ea9b90-2228-43b0-9552-725274cb7c67",
   "metadata": {},
   "source": [
    "现在计算y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "1f8f6bd1-5e57-4737-b812-76eb2610eb18",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(28., grad_fn=<MulBackward0>)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = 2 * torch.dot(x,x)\n",
    "y"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "05724c19-69e0-4dd5-8e6d-4736be22027c",
   "metadata": {},
   "source": [
    "通过调用反响传播函数来自动计算y关于x每个分量的梯度"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "e3152030-c811-4ca3-93f0-b6824d9532c3",
   "metadata": {},
   "outputs": [],
   "source": [
    "y.backward()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "23c623db-216c-4ae9-be6e-e1fa1f91d4ff",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([ 0.,  4.,  8., 12.])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.grad"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "ec93f249-da1d-4a85-9f71-3111d913f6ee",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([True, True, True, True])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.grad == 4 * x"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "48d3ad10-30dd-4364-9c8a-fa05d217fe6f",
   "metadata": {},
   "source": [
    "现在计算x的另一个函数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "b3e27e50-80ac-4d62-9ff5-8d60745b1279",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([0., 0., 0., 0.])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 在默认情况下，PyTorch会累积梯度，我们需要清除之前的值\n",
    "x.grad.zero_()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "f463fedd-0dac-40ce-a822-a94030cf9984",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(tensor([0., 1., 2., 3.], requires_grad=True),\n",
       " tensor(6., grad_fn=<SumBackward0>))"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = x.sum()\n",
    "x,y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "ef896573-4944-4386-9d45-f11f28fb62e5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(6., grad_fn=<SumBackward0>)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y.backward()\n",
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "9c2de676-1e7c-4e60-a7bc-71d8e63c8a54",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([1., 1., 1., 1.])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.grad"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "345dd5c4-377f-4584-b45d-e50435d6c241",
   "metadata": {},
   "source": [
    "在深度学习中，我们的目标不是计算微分矩阵，而是对批量中每个样本进行单独计算偏导数，并将它们的和作为最终结果。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "65393e36-4518-4401-b52f-eafb22711df7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(tensor([0., 1., 2., 3.], requires_grad=True),\n",
       " tensor([0., 1., 4., 9.], grad_fn=<MulBackward0>),\n",
       " tensor([0., 2., 4., 6.]))"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.grad.zero_()\n",
    "y = x * x\n",
    "# 等价于 y.backward(torch.ones(len(x)))\n",
    "y.sum().backward()\n",
    "x,y,x.grad"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "36012c56-1c91-480f-8543-1500a4389b61",
   "metadata": {},
   "source": [
    "将某些计算移动到记录的计算图之外"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1abbf388-39bb-43d4-9004-ad62ec41cc01",
   "metadata": {},
   "source": [
    "detach() 方法用于返回一个新的 Tensor，这个 Tensor 和原来的 Tensor 共享相同的内存空间，但是不会被计算图所追踪，也就是说它不会参与反向传播，不会影响到原有的计算图，这使得它成为处理中间结果的一种有效方式\n",
    "\n",
    "https://zhuanlan.zhihu.com/p/613991185"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "a6683e33-6793-4ac1-ba5c-1eba55e190de",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(tensor([True, True, True, True]),\n",
       " tensor([0., 1., 4., 9.]),\n",
       " tensor([0., 1., 4., 9.]),\n",
       " tensor([0., 1., 4., 9.], grad_fn=<MulBackward0>))"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.grad.zero_()\n",
    "y = x * x\n",
    "u = y.detach()\n",
    "z = u * x\n",
    "\n",
    "z.sum().backward()\n",
    "x.grad == u, x.grad, u,y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "d13beaa1-2575-4bfd-82f3-e2f704657a4f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(tensor([True, True, True, True]),\n",
       " tensor([0., 1., 2., 3.], requires_grad=True),\n",
       " tensor([0., 1., 4., 9.], grad_fn=<MulBackward0>))"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.grad.zero_()\n",
    "y.sum().backward()\n",
    "x.grad == 2 *x, x, y"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ae222887-26d4-463c-9496-e52ef22b00eb",
   "metadata": {},
   "source": [
    "构建函数的计算图，如通过python控制流（条件、循环或任意函数调用），我们仍然可以计算的到变量的梯度。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "837e30b7-5d87-44c7-94ff-e73d8743efcc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(True)"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(a):\n",
    "    b = a *2\n",
    "    while b.norm() < 100:\n",
    "        b = b *2\n",
    "    if b.sum() > 0:\n",
    "        c = b\n",
    "    else:\n",
    "        c = 100 * b\n",
    "    return c\n",
    "\n",
    "a = torch.randn(size=(), requires_grad=True)\n",
    "d = f(a)\n",
    "d.backward()\n",
    "\n",
    "a.grad == d / a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "9725bb9d-bdae-4949-a34b-15ae058a1e42",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(True)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 导入PyTorch库（尽管这在代码中没有明确显示，但根据上下文可以推断出）  \n",
    "import torch  \n",
    "  \n",
    "# 定义一个函数f，接受一个参数a  \n",
    "def f(a):  \n",
    "    # 将a的值乘以2，并赋值给b  \n",
    "    b = a * 2  \n",
    "      \n",
    "    # 当b的范数小于100时，执行循环体  \n",
    "    while b.norm() < 100:  \n",
    "        # 将b的值乘以2  \n",
    "        b = b * 2  \n",
    "      \n",
    "    # 判断b的元素之和是否大于0  \n",
    "    if b.sum() > 0:  \n",
    "        # 如果大于0，将b的值赋给c  \n",
    "        c = b  \n",
    "    else:  \n",
    "        # 如果小于或等于0，将b的值乘以100后赋给c  \n",
    "        c = 100 * b  \n",
    "      \n",
    "    # 返回c的值  \n",
    "    return c  \n",
    "  \n",
    "# 创建一个随机的PyTorch张量a，该张量可计算梯度  \n",
    "a = torch.randn(size=(), requires_grad=True)  \n",
    "  \n",
    "# 调用函数f，并将结果赋值给d  \n",
    "d = f(a)  \n",
    "  \n",
    "# 对d进行反向传播，计算梯度  \n",
    "d.backward()  \n",
    "  \n",
    "# 检查计算得到的梯度是否等于d / a，并返回结果  \n",
    "a.grad == d / a"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "87897985-12b3-4497-bd4f-325b439359c6",
   "metadata": {},
   "source": [
    "size=()表示生成的张量（tensor）的形状。具体来说，size=()意味着生成一个标量（scalar），也就是一个0维的张量。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "757f3699-4fa0-4293-bb50-9a29669a9640",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ec42c2ef-1ae9-4222-9e61-965a30746455",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f5405a9a-d0f6-40b5-8af0-632414012b4b",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "cba73500-077f-4fdd-b569-28b056dfedb7",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ac403358-bd09-44f8-b58d-4012e23f6253",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6278f3e5-b870-48fe-8cb1-78834c7f567f",
   "metadata": {},
   "outputs": [],
   "source": []
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  {
   "cell_type": "code",
   "execution_count": null,
   "id": "442eb3f2-1d27-4b59-9f63-dbe1705fdabc",
   "metadata": {},
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   "source": []
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   "cell_type": "code",
   "execution_count": null,
   "id": "07ecd417-e871-4fc0-8358-cfaa75f04ec1",
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   "cell_type": "code",
   "execution_count": null,
   "id": "bbebcb6c-0b14-473d-9d2c-30806820dd24",
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   "cell_type": "code",
   "execution_count": null,
   "id": "6c0f047e-c4ad-4030-9d4b-20adc8d5be45",
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