code/64-注意力机制.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "YH8iJl3qtk1a"
      },
      "source": [
        "#64-注意力机制"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "NUoif5LqtOMh"
      },
      "source": [
        "##启用GPU\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "slNovdubtU8A",
        "outputId": "b898f90e-3a89-4db0-a09b-e021ec110697"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Found GPU at: /device:GPU:0\n"
          ]
        }
      ],
      "source": [
        "%tensorflow_version 2.x\n",
        "import tensorflow as tf\n",
        "device_name = tf.test.gpu_device_name()\n",
        "if device_name != '/device:GPU:0':\n",
        "  raise SystemError('GPU device not found')\n",
        "print('Found GPU at: {}'.format(device_name))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ZdGON_oxtY34"
      },
      "source": [
        "##安装pytorch和d2l"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "shMeiKRmtZe4",
        "outputId": "4b905b3c-59be-4c78-ab80-63e1c062bebd"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Requirement already satisfied: torch in /usr/local/lib/python3.7/dist-packages (1.10.0+cu111)\n",
            "Requirement already satisfied: torchvision in /usr/local/lib/python3.7/dist-packages (0.11.1+cu111)\n",
            "Requirement already satisfied: torchaudio in /usr/local/lib/python3.7/dist-packages (0.10.0+cu111)\n",
            "Requirement already satisfied: typing-extensions in /usr/local/lib/python3.7/dist-packages (from torch) (3.10.0.2)\n",
            "Requirement already satisfied: numpy in /usr/local/lib/python3.7/dist-packages (from torchvision) (1.19.5)\n",
            "Requirement already satisfied: pillow!=8.3.0,>=5.3.0 in /usr/local/lib/python3.7/dist-packages (from torchvision) (7.1.2)\n"
          ]
        }
      ],
      "source": [
        "!pip3 install torch torchvision torchaudio"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "HakkP1iKtaBH",
        "outputId": "22f43a5e-b756-4b65-b11c-3a495d7a6dcf"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Collecting d2l==0.14\n",
            "  Downloading d2l-0.14.0-py3-none-any.whl (48 kB)\n",
            "\u001b[K     |████████████████████████████████| 48 kB 2.7 MB/s \n",
            "\u001b[?25hRequirement already satisfied: matplotlib in /usr/local/lib/python3.7/dist-packages (from d2l==0.14) (3.2.2)\n",
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            "Requirement already satisfied: pyrsistent!=0.17.0,!=0.17.1,!=0.17.2,>=0.14.0 in /usr/local/lib/python3.7/dist-packages (from jsonschema!=2.5.0,>=2.4->nbformat>=4.2.0->ipywidgets->jupyter->d2l==0.14) (0.18.1)\n",
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            "Requirement already satisfied: importlib-resources>=1.4.0 in /usr/local/lib/python3.7/dist-packages (from jsonschema!=2.5.0,>=2.4->nbformat>=4.2.0->ipywidgets->jupyter->d2l==0.14) (5.4.0)\n",
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            "Installing collected packages: d2l\n",
            "Successfully installed d2l-0.14.0\n"
          ]
        }
      ],
      "source": [
        "!pip3 install d2l==0.14"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "853xPsHhptNV"
      },
      "source": [
        "###注意力的可视化\n",
        "\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "K7eqrHFes6-W"
      },
      "outputs": [],
      "source": [
        "import torch\n",
        "from d2l import torch as d2l"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "UeJD_VVb3azN"
      },
      "source": [
        "\n",
        "\n",
        "*   可视化注意力的权重[0,1]\n",
        "*   输入matrices的形状：要显示的行数，要显示的列数，查询的数目，键的数目\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "OoBxfYpFuC1u"
      },
      "outputs": [],
      "source": [
        "def show_heatmaps(matrices, xlabel, ylabel, titles=None, figsize=(2.5, 2.5),\n",
        "                  cmap='Reds'):\n",
        "    \"\"\"显示矩阵热图\"\"\"\n",
        "    d2l.use_svg_display()\n",
        "    num_rows, num_cols = matrices.shape[0], matrices.shape[1]\n",
        "    fig, axes = d2l.plt.subplots(num_rows, num_cols, figsize=figsize,\n",
        "                                 sharex=True, sharey=True, squeeze=False)\n",
        "    for i, (row_axes, row_matrices) in enumerate(zip(axes, matrices)):\n",
        "        for j, (ax, matrix) in enumerate(zip(row_axes, row_matrices)):\n",
        "            pcm = ax.imshow(matrix.detach().numpy(), cmap=cmap)\n",
        "            if i == num_rows - 1:\n",
        "                ax.set_xlabel(xlabel)\n",
        "            if j == 0:\n",
        "                ax.set_ylabel(ylabel)\n",
        "            if titles:\n",
        "                ax.set_title(titles[j])\n",
        "    fig.colorbar(pcm, ax=axes, shrink=0.6)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "1XXPEZ8iwOQL"
      },
      "source": [
        "概率矩阵演示。仅当查询和键相同时，注意力权重为1，否则为0\n",
        "\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 229
        },
        "id": "MIdsrzecwLAO",
        "outputId": "1357ccb1-b285-4e8f-e711-aba6b6864113"
      },
      "outputs": [
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            "text/plain": [
              "<Figure size 180x180 with 2 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "attention_weights = torch.eye(10).reshape((1, 1, 10, 10))\n",
        "show_heatmaps(attention_weights, xlabel='Keys', ylabel='Queries')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "XtPmjBOus5Fp"
      },
      "source": [
        "##注意力汇聚：Nadaraya-Watson核回归"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "aGc2DVS6wJtI"
      },
      "outputs": [],
      "source": [
        "import torch\n",
        "from torch import nn\n",
        "from d2l import torch as d2l"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "BYx95qIS5dLs"
      },
      "source": [
        "###生成数据集\n",
        "\n",
        "根据非线性函数：\\begin{equation} y_i = 2sin(x_i) + x_{i}^{0.8} + \\xi \\end{equation}\n",
        "生成数据集"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "01_GUkxdz0Tg"
      },
      "outputs": [],
      "source": [
        "n_train = 50 # 训练样本数\n",
        "x_train, _ = torch.sort(torch.rand(n_train)*5) # 排序后的训练样本"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "ObpH0AgS0SnX",
        "outputId": "76f11d43-a9c5-403b-825f-5fd4fed24ebe"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "50"
            ]
          },
          "execution_count": 22,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "def f(x):\n",
        "    return 2 * torch.sin(x) + x**0.8\n",
        "\n",
        "y_train = f(x_train) + torch.normal(0.0,0.5,(n_train,)) # 训练样本的输出\n",
        "x_test = torch.arange(0,5,0.1) # 测试样本\n",
        "y_truth = f(x_test) # 测试样本的真实输出\n",
        "n_test = len(x_test) # 测试样本数\n",
        "n_test"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "PATRJgPR7nhh"
      },
      "source": [
        "\n",
        "\n",
        "*   训练样本（由圆圈表示）\n",
        "*   不带噪声项的真实数据生成的函数f（标记为\"Truth\"）\n",
        "*   学习得到的预测函数（标记为\"Pred\"）\n",
        "\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "dadBcSmY1Clh"
      },
      "outputs": [],
      "source": [
        "def plot_kernel_reg(y_hat):\n",
        "    d2l.plot(x_test,[y_truth,y_hat],'x','y',legend=['Truth','Pred'],xlim=[0,5],ylim=[-1,5])\n",
        "    d2l.plt.plot(x_train,y_train,'o',alpha=0.5)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "eui1-By52VL_"
      },
      "source": [
        "###平均汇聚\n",
        "平均汇聚的公式\\begin{equation} f(x) = \\frac{1}{n}Σ_{i=1}^{n}y_i\\end{equation}"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 267
        },
        "id": "eM5YxumL1ujt",
        "outputId": "221d8c51-4608-4061-ecd4-6ceaee780027"
      },
      "outputs": [
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            "text/plain": [
              "<Figure size 252x180 with 1 Axes>"
            ]
          },
          "metadata": {
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        }
      ],
      "source": [
        "y_hat = torch.repeat_interleave(y_train.mean(),n_test)\n",
        "plot_kernel_reg(y_hat)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "wUlibgnD2a0k"
      },
      "source": [
        "###非参数注意力汇聚\n",
        "由于平均汇聚忽略了输入，因此根据输入的位置对输出进行加权：\n",
        "\\begin{equation} f(x)=Σ_{i=1}^{n}\\frac{K(x-x_i)}{Σ_{i=1}^{n}K(x-x_i)}y_i \\end{equation}\n",
        "经过改良，获得更通用的注意力汇聚公式：\n",
        "\\begin{equation} f(x)=Σ_{i=1}^{n}α(x,x_i)y_i \\end{equation}\n",
        "令高斯核为：\n",
        "\\begin{equation} K(u)=\\frac{1}{\\sqrt{2π}}exp(-\\frac{u^2}{2}) \\end{equation} \n",
        "将高斯核带入注意力汇聚公式：\n",
        "\\begin{equation} f(x)=Σ_{i=1}^{n}softmax(-\\frac{1}{2}(x-x_i)^2)y_i \\end{equation}\n",
        "如果一个键越是接近给定的查询，那么分配给这个键对应值的注意力权重就会越大，也就是“获得了更多的注意力”"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
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        },
        "id": "9-e7Bhce17vq",
        "outputId": "a076cd95-acba-485b-84ff-98e383e7216d"
      },
      "outputs": [
        {
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      ],
      "source": [
        "# X_repeat的形状：（n_test,n_train）,\n",
        "# 每一行都包含着相同的测试输入（例如：同样的查询）\n",
        "X_repeat = x_test.repeat_interleave(n_train).reshape((-1,n_train))\n",
        "# x_train包含着键。attention_weights的形状：（n_test,n_train）,\n",
        "# 每一行都包含着要在给定的每个查询的值（y_train）之间分配的注意力权重\n",
        "attention_weights = nn.functional.softmax(-(X_repeat-x_train)**2 / 2,dim=1)\n",
        "# y_hat的每个元素都是值的加权平均值，其中的权重是注意力权重\n",
        "y_hat = torch.matmul(attention_weights,y_train)\n",
        "plot_kernel_reg(y_hat)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "7C77gUJmBn6u"
      },
      "source": [
        "\n",
        "*   测试数据的输入相当于查询\n",
        "*   训练数据的输入相当于键\n",
        "\n",
        "“查询——键”对越接近，注意力汇聚的注意力权重越高\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 232
        },
        "id": "mU8NV5Yp2_r7",
        "outputId": "dafd509a-789a-44b6-96f4-a79b68738ee7"
      },
      "outputs": [
        {
          "data": {
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            "text/plain": [
              "<Figure size 180x180 with 2 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "show_heatmaps(attention_weights.unsqueeze(0).unsqueeze(0),xlabel='Sorted training inputs',ylabel='Sorted testing inputs')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "TOJQBe9p3uPw"
      },
      "source": [
        "###带参数注意力汇聚\n",
        "\n",
        "\\begin{equation} f(x)=Σ_{i=1}^{n}softmax(-\\frac{1}{2}((x-x_i)\\omega)^2)y_i \\end{equation}\n",
        "\n",
        "*   批量矩阵乘法\n",
        "\n",
        "\\begin{equation} (n,a,b),(n,b,c) \\to (n,a,c) \\end{equation}\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "0SHYywlG3XRb",
        "outputId": "7fd3eba5-338a-44f9-e2ad-53352771b42c"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "torch.Size([2, 1, 6])"
            ]
          },
          "execution_count": 29,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "X = torch.ones((2,1,4))\n",
        "Y = torch.ones((2,4,6))\n",
        "torch.bmm(X,Y).shape"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "AqYuAVbGC2OJ"
      },
      "source": [
        "计算加权平均值"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "fCYBwGT54IOq",
        "outputId": "09293569-4798-4595-9f6c-9d2881dd0aa2"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "tensor([[[ 4.5000]],\n",
              "\n",
              "        [[14.5000]]])"
            ]
          },
          "execution_count": 31,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "weights = torch.ones((2,10)) * 0.1\n",
        "values = torch.arange(20.0).reshape((2,10))\n",
        "torch.bmm(weights.unsqueeze(1),values.unsqueeze(-1))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "JyZSOiee4PSu"
      },
      "source": [
        "*   定义模型"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "Kr9S9BPm5K2U"
      },
      "outputs": [],
      "source": [
        "class NWKernelRegression(nn.Module):\n",
        "    def __init__(self,**kwargs):\n",
        "        super().__init__(**kwargs)\n",
        "        self.w = nn.Parameter(torch.rand((1,),requires_grad=True))\n",
        "\n",
        "    def forward(self,queries,keys,values):\n",
        "        # queries和attention_weights的形状为(查询个数，\"键——值\"对个数)\n",
        "        queries = queries.repeat_interleave(keys.shape[1]).reshape((-1,keys.shape[1]))\n",
        "        self.attention_weights = nn.functional.softmax(-((queries - keys) * self.w)**2 /2,dim=1)\n",
        "        # values的形状为（查询个数，\"键——值\"对个数）\n",
        "        return torch.bmm(self.attention_weights.unsqueeze(1),values.unsqueeze(-1)).reshape(-1)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "xgS-g5fk63_t"
      },
      "source": [
        "*   训练"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "XQ0l0wPO6_BM"
      },
      "outputs": [],
      "source": [
        "# X_title的形状：（n_train，n_train），每一行都包含着相同的训练输入\n",
        "X_tile = x_train.repeat((n_train,1))\n",
        "# Y_title的形状：（n_train，n_train），每一行都包含着相同的训练输入\n",
        "Y_tile = y_train.repeat((n_train,1))\n",
        "# keys的形状：（'n_train'，'n_train'-1）\n",
        "keys = X_tile[(1 - torch.eye(n_train)).type(torch.bool)].reshape((n_train,-1))\n",
        "# values的形状：（'n_train'，'n_train'-1）\n",
        "values = Y_tile[(1 - torch.eye(n_train)).type(torch.bool)].reshape((n_train,-1))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "PyHIJdEqERRI"
      },
      "source": [
        "使用平方损失函数和随机梯度下降"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 262
        },
        "id": "SrL-oq1D79Ao",
        "outputId": "c5668575-5c96-480b-9056-7020e7aa298a"
      },
      "outputs": [
        {
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            "text/plain": [
              "<Figure size 252x180 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "net = NWKernelRegression()\n",
        "loss = nn.MSELoss(reduction='none')\n",
        "trainer = torch.optim.SGD(net.parameters(),lr=0.5)\n",
        "animator = d2l.Animator(xlabel='epoch',ylabel='loss',xlim=[1,5])\n",
        "\n",
        "for epoch in range(5):\n",
        "    trainer.zero_grad()\n",
        "    l = loss(net(x_train,keys,values),y_train)\n",
        "    l.sum().backward()\n",
        "    trainer.step()\n",
        "    print(f'epoch{epoch + 1}, loss{float(l.sum()):.6f}')\n",
        "    animator.add(epoch + 1,float(l.sum()))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "dwPrUM1TEaew"
      },
      "source": [
        "带噪声的数据训练结果不如非参数模型的平滑"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 267
        },
        "id": "Ka6ok93A9ty-",
        "outputId": "c4ef5c8b-67e4-493e-a869-59c1328810a0"
      },
      "outputs": [
        {
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            "text/plain": [
              "<Figure size 252x180 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "# keys的形状：（n_test，n_train），每一行包含着相同的训练输入（eg.相同的键）\n",
        "keys = x_train.repeat((n_test,1))\n",
        "# value的形状：（n_test，n_train)\n",
        "values = y_train.repeat((n_test,1))\n",
        "y_hat = net(x_test,keys,values).unsqueeze(1).detach()\n",
        "plot_kernel_reg(y_hat)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 232
        },
        "id": "aI43gX-tAT1h",
        "outputId": "5af9eda8-b035-4493-8fb9-689978a8b5da"
      },
      "outputs": [
        {
          "data": {
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            "text/plain": [
              "<Figure size 180x180 with 2 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "show_heatmaps(net.attention_weights.unsqueeze(0).unsqueeze(0),xlabel='Sorted training inputs',ylabel='Sorted testing input')"
      ]
    }
  ],
  "metadata": {
    "accelerator": "GPU",
    "colab": {
      "name": "Attention.ipynb",
      "provenance": []
    },
    "kernelspec": {
      "display_name": "Python 3",
      "name": "python3"
    },
    "language_info": {
      "name": "python"
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  "nbformat": 4,
  "nbformat_minor": 0
}