src/10-2.ipynb

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  {
   "cell_type": "markdown",
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   "source": [
    "## 10.2 注意力汇聚：Nadaraya-Watson核回归"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import torch\n",
    "from torch import nn\n",
    "from d2l import torch as d2l"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 10.2.1 生成数据集"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "根据下面的非线性函数生成一个人工数据集，其中加入的噪声为$\\epsilon$\n",
    "\n",
    "$$\n",
    "y_i=2\\sin(x_i)+x_i^{0.8}+\\epsilon\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "50"
      ]
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     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "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\n",
    "\n",
    "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": {},
   "source": [
    "下面的函数将绘制所有的训练样本、不带噪声项的真实数据、以及学到的预测函数。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_kernel_reg(y_hat):\n",
    "    d2l.plot(x_test, [y_truth, y_hat], 'x', 'y', legend=['Truth', 'Pred'],\n",
    "             xlim=[0, 5], ylim=[-1, 5])\n",
    "    d2l.plt.plot(x_train, y_train, 'o', alpha=0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 10.2.2 平均汇聚"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "先使用一个最简单的估计器来估计\n",
    "\n",
    "$$\n",
    "f(x)=\\frac{1}{n}\\sum_{i=1}^ny_i\n",
    "$$\n",
    "\n",
    "但是很明显这种平均数的方法很不聪明"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
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    "y_hat = torch.repeat_interleave(y_train.mean(), n_test)\n",
    "plot_kernel_reg(y_hat)"
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   "source": [
    "### 10.2.3 非参数注意力汇聚"
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   "source": [
    "Nadaraya和Watson提出了一种更好的想法，也就是根据输入的位置对$y_i$进行加权：\n",
    "\n",
    "$$\n",
    "f(x)=\\sum_{i=1}^n\\frac{K(x-x_i)}{\\sum_{j=1}^nK(x-x_j)}y_i\n",
    "$$\n",
    "\n",
    "其中的$K$就是核。上面的公式可以转化为一个更为通用的**注意力汇聚(attention pooling)**公式：\n",
    "\n",
    "$$\n",
    "f(x)=\\sum_{i=1}^n\\alpha(x,x_i)y_i\n",
    "$$\n",
    "\n",
    "这里的$x$是查询$(x_i,y_i)$是键值对，将查询$x$和键$x_i$之间的建模关系为**注意力权重(attention weight)** $\\alpha(x,x_i)$，这个权重将被分配给每一个对应值$y_i$。对于任何查询，模型在所有键值对注意力权重都是一个有效的概率分布，也就是所有都是非负的，并且总和为1。\n",
    "\n",
    "下面是一个高斯核(Gaussian kernel):\n",
    "\n",
    "$$\n",
    "K(u)=\\frac{1}{\\sqrt{2\\pi}}\\exp\\left(-\\frac{u^2}{2}\\right)\n",
    "$$\n",
    "\n",
    "带入计算可得函数$f(x)$\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "f(x)&=\\sum_{i=1}^n\\alpha(x,x_i)y_i\\\\\n",
    "&=\\sum_{i=1}^n\\frac{\\exp\\left(-\\frac{1}{2}(x-x_i)^2\\right)}{\\sum_{j=1}^n\\exp\\left(-\\frac{1}{2}(x-x_j)^2\\right)}y_i\\\\\n",
    "&=\\sum_{i=1}^n\\mathrm{softmax}\\left(-\\frac{1}{2}(x-x_i)^2\\right)y_i\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "也就是说，如果一个键越是接近给定的查询$x$，那么分配给这个键对应值$y_i$的注意力就会越大。\n",
    "\n",
    "需要注意的是，这是一个没有参数的模型，也就是**非参数的注意力汇聚**模型。"
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  {
   "cell_type": "code",
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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",
    "# X_repeat.shape=[n_test, n_train]\n",
    "# x_train.shape=[n_train]\n",
    "# X_repeat-x_train就是每一行独自减去x_train\n",
    "# 在这里实际上已经是进行了查询了\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": {},
   "source": [
    "上述代码中的矩阵大小变化如下所示。\n",
    "\n",
    "![](../images/10-2-a1.svg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "<Figure size 250x250 with 2 Axes>"
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     },
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    }
   ],
   "source": [
    "d2l.show_heatmaps(attention_weights.unsqueeze(0).unsqueeze(0),\n",
    "                  xlabel='Sorted training inputs',\n",
    "                  ylabel='Sorted testing inputs')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 10.2.4 带参数的注意力汇聚"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "也就是在查询$x$和键$x_i$之间的距离乘了一个可学习的参数$w$：\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "f(x)&=\\sum_{i=1}^n\\alpha(x,x_i)y_i\\\\\n",
    "&=\\sum_{i=1}^n\\mathrm{softmax}\\left(-\\frac{1}{2}((x-x_i)w)^2\\right)y_i\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 10.2.4.1 批量矩阵乘法"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "也就是对于两个批次数量相同的矩阵，分别在每个每个批次上进行矩阵乘法，也就是批次的维度不变，然后后面的两个维度执行矩阵乘法。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "torch.Size([2, 1, 6])"
      ]
     },
     "execution_count": 7,
     "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": {},
   "source": [
    "在注意力机制的背景中，可以使用小批量矩阵乘法来计算小批量数据中的加权平均值。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "torch.Size([2, 1, 10]) torch.Size([2, 10, 1])\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "tensor([[[ 4.5000]],\n",
       "\n",
       "        [[14.5000]]])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "weights = torch.ones((2, 10)) * 0.1\n",
    "values = torch.arange(20.0).reshape((2, 10))\n",
    "print(weights.unsqueeze(1).shape, values.unsqueeze(-1).shape)\n",
    "torch.bmm(weights.unsqueeze(1), values.unsqueeze(-1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 10.2.4.2 定义模型"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "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",
    "    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(\n",
    "            -((queries - keys) * self.w) ** 2 / 2,\n",
    "            dim=1,\n",
    "        )\n",
    "        return torch.bmm(\n",
    "            self.attention_weights.unsqueeze(1),\n",
    "            values.unsqueeze(-1),\n",
    "        ).reshape(-1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 10.2.4.3 训练"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(torch.Size([50, 50]),\n",
       " torch.Size([50, 50]),\n",
       " torch.Size([50, 49]),\n",
       " torch.Size([50, 49]))"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# X_tile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_tile的形状:(n_train，n_train)，每一行都包含着相同的训练输出\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))\n",
    "\n",
    "X_tile.shape, Y_tile.shape, keys.shape, values.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "         [0.3506, 0.6771, 0.8726,  ..., 4.6137, 4.7151, 4.7647],\n",
       "         [0.3506, 0.6771, 0.8726,  ..., 4.6137, 4.7151, 4.7647],\n",
       "         ...,\n",
       "         [0.3506, 0.6771, 0.8726,  ..., 4.6137, 4.7151, 4.7647],\n",
       "         [0.3506, 0.6771, 0.8726,  ..., 4.6137, 4.7151, 4.7647],\n",
       "         [0.3506, 0.6771, 0.8726,  ..., 4.6137, 4.7151, 4.7647]]),\n",
       " tensor([[0.8104, 2.5698, 2.3593,  ..., 0.9263, 1.1831, 0.7722],\n",
       "         [0.8104, 2.5698, 2.3593,  ..., 0.9263, 1.1831, 0.7722],\n",
       "         [0.8104, 2.5698, 2.3593,  ..., 0.9263, 1.1831, 0.7722],\n",
       "         ...,\n",
       "         [0.8104, 2.5698, 2.3593,  ..., 0.9263, 1.1831, 0.7722],\n",
       "         [0.8104, 2.5698, 2.3593,  ..., 0.9263, 1.1831, 0.7722],\n",
       "         [0.8104, 2.5698, 2.3593,  ..., 0.9263, 1.1831, 0.7722]]),\n",
       " tensor([[0.6771, 0.8726, 0.9297,  ..., 4.6137, 4.7151, 4.7647],\n",
       "         [0.3506, 0.8726, 0.9297,  ..., 4.6137, 4.7151, 4.7647],\n",
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       "         ...,\n",
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       "         [0.3506, 0.6771, 0.8726,  ..., 4.5722, 4.6137, 4.7151]]),\n",
       " tensor([[2.5698, 2.3593, 1.9499,  ..., 0.9263, 1.1831, 0.7722],\n",
       "         [0.8104, 2.3593, 1.9499,  ..., 0.9263, 1.1831, 0.7722],\n",
       "         [0.8104, 2.5698, 1.9499,  ..., 0.9263, 1.1831, 0.7722],\n",
       "         ...,\n",
       "         [0.8104, 2.5698, 2.3593,  ..., 1.4396, 1.1831, 0.7722],\n",
       "         [0.8104, 2.5698, 2.3593,  ..., 1.4396, 0.9263, 0.7722],\n",
       "         [0.8104, 2.5698, 2.3593,  ..., 1.4396, 0.9263, 1.1831]]))"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_tile, Y_tile, keys, values"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "下图展示了其在制作`keys`和`values`时候所进行的技巧。\n",
    "\n",
    "![](../images/10-2-a2.svg)\n",
    "\n",
    "但是非常不能够理解这样做的目的是什么。\n",
    "\n",
    "下面使用平方损失函数和随机梯度下降进行训练。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
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     "metadata": {},
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   ],
   "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": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
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   ],
   "source": [
    "# keys的形状:(n_test，n_train)，每一行包含着相同的训练输入（例如，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": 14,
   "metadata": {},
   "outputs": [
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   "source": [
    "d2l.show_heatmaps(net.attention_weights.unsqueeze(0).unsqueeze(0),\n",
    "                  xlabel='Sorted training inputs',\n",
    "                  ylabel='Sorted testing inputs')"
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   "source": [
    "### 10.2.5 小结"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Nadaraya-Watson核回归是具有注意力机制的机器学习范例。\n",
    "- Nadaraya-Watson核回归的注意力汇聚是对训练数据中输出的加权平均。从注意力的角度来看，分配给每个值的注意力权重取决于将值所对应的键和查询作为输入的函数。\n",
    "- 注意力汇聚可以分为非参数型和带参数型。"
   ]
  }
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