5681 lines
244 KiB
Plaintext
5681 lines
244 KiB
Plaintext
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"source": [
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"# 注意力汇聚：Nadaraya-Watson 核回归\n",
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":label:`sec_nadaraya-watson`\n",
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"\n",
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"上节介绍了框架下的注意力机制的主要成分 :numref:`fig_qkv`：\n",
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"查询（自主提示）和键（非自主提示）之间的交互形成了注意力汇聚；\n",
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"注意力汇聚有选择地聚合了值（感官输入）以生成最终的输出。\n",
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"本节将介绍注意力汇聚的更多细节，\n",
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"以便从宏观上了解注意力机制在实践中的运作方式。\n",
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"具体来说，1964年提出的Nadaraya-Watson核回归模型\n",
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"是一个简单但完整的例子，可以用于演示具有注意力机制的机器学习。\n"
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]
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"cell_type": "code",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T07:07:14.699824Z",
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},
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"origin_pos": 2,
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"tab": [
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"pytorch"
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]
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},
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"outputs": [],
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"source": [
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"import torch\n",
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"from torch import nn\n",
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"from d2l import torch as d2l"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"source": [
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"## [**生成数据集**]\n",
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"\n",
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"简单起见，考虑下面这个回归问题：\n",
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"给定的成对的“输入－输出”数据集\n",
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"$\\{(x_1, y_1), \\ldots, (x_n, y_n)\\}$，\n",
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"如何学习$f$来预测任意新输入$x$的输出$\\hat{y} = f(x)$？\n",
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"\n",
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"根据下面的非线性函数生成一个人工数据集，\n",
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"其中加入的噪声项为$\\epsilon$：\n",
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"\n",
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"$$y_i = 2\\sin(x_i) + x_i^{0.8} + \\epsilon,$$\n",
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"\n",
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"其中$\\epsilon$服从均值为$0$和标准差为$0.5$的正态分布。\n",
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"在这里生成了$50$个训练样本和$50$个测试样本。\n",
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"为了更好地可视化之后的注意力模式，需要将训练样本进行排序。\n"
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]
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"cell_type": "code",
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"execution_count": 2,
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"id": "77ea63dd",
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"metadata": {
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"execution": {
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"pytorch"
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"outputs": [],
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"source": [
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"n_train = 50 # 训练样本数\n",
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"x_train, _ = torch.sort(torch.rand(n_train) * 5) # 排序后的训练样本"
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]
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},
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"pytorch"
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"data": {
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"text/plain": [
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"50"
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]
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
|
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"def f(x):\n",
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" return 2 * torch.sin(x) + x**0.8\n",
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"\n",
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"y_train = f(x_train) + torch.normal(0.0, 0.5, (n_train,)) # 训练样本的输出\n",
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"x_test = torch.arange(0, 5, 0.1) # 测试样本\n",
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"y_truth = f(x_test) # 测试样本的真实输出\n",
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"n_test = len(x_test) # 测试样本数\n",
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"n_test"
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]
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},
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"cell_type": "markdown",
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"source": [
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"下面的函数将绘制所有的训练样本（样本由圆圈表示），\n",
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"不带噪声项的真实数据生成函数$f$（标记为“Truth”），\n",
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"以及学习得到的预测函数（标记为“Pred”）。\n"
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]
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},
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"cell_type": "code",
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"execution_count": 4,
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"id": "84166e26",
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"origin_pos": 15,
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"tab": [
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"pytorch"
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]
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},
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"outputs": [],
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"source": [
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"def plot_kernel_reg(y_hat):\n",
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" d2l.plot(x_test, [y_truth, y_hat], 'x', 'y', legend=['Truth', 'Pred'],\n",
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" xlim=[0, 5], ylim=[-1, 5])\n",
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" d2l.plt.plot(x_train, y_train, 'o', alpha=0.5);"
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]
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},
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{
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"cell_type": "markdown",
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"id": "83acd3e5",
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"metadata": {
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},
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"source": [
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"## 平均汇聚\n",
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"\n",
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"先使用最简单的估计器来解决回归问题。\n",
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"基于平均汇聚来计算所有训练样本输出值的平均值：\n",
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"\n",
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||
"$$f(x) = \\frac{1}{n}\\sum_{i=1}^n y_i,$$\n",
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||
":eqlabel:`eq_avg-pooling`\n",
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"\n",
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"如下图所示，这个估计器确实不够聪明。\n",
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"真实函数$f$（“Truth”）和预测函数（“Pred”）相差很大。\n"
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]
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},
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"origin_pos": 18,
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"## [**非参数注意力汇聚**]\n",
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"\n",
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"\n",
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"\n",
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"\n",
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|
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"比较 :eqref:`eq_attn-pooling`和 :eqref:`eq_avg-pooling`，\n",
|
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|
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|
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"\n",
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"\n",
|
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|
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|
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"\n",
|
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"$$\\begin{aligned} f(x) &=\\sum_{i=1}^n \\alpha(x, x_i) y_i\\\\ &= \\sum_{i=1}^n \\frac{\\exp\\left(-\\frac{1}{2}(x - x_i)^2\\right)}{\\sum_{j=1}^n \\exp\\left(-\\frac{1}{2}(x - x_j)^2\\right)} y_i \\\\&= \\sum_{i=1}^n \\mathrm{softmax}\\left(-\\frac{1}{2}(x - x_i)^2\\right) y_i. \\end{aligned}$$\n",
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"\n",
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|
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"那么分配给这个键对应值$y_i$的注意力权重就会越大，\n",
|
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"\n",
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"值得注意的是，Nadaraya-Watson核回归是一个非参数模型。\n",
|
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"因此， :eqref:`eq_nadaraya-watson-gaussian`是\n",
|
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"*非参数的注意力汇聚*（nonparametric attention pooling）模型。\n",
|
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|
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"从绘制的结果会发现新的模型预测线是平滑的，并且比平均汇聚的预测更接近真实。\n"
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|
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"## [**带参数注意力汇聚**]\n",
|
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"\n",
|
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"非参数的Nadaraya-Watson核回归具有*一致性*（consistency）的优点：\n",
|
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|
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|
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"\n",
|
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|
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|
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"\n",
|
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"$$\\begin{aligned}f(x) &= \\sum_{i=1}^n \\alpha(x, x_i) y_i \\\\&= \\sum_{i=1}^n \\frac{\\exp\\left(-\\frac{1}{2}((x - x_i)w)^2\\right)}{\\sum_{j=1}^n \\exp\\left(-\\frac{1}{2}((x - x_j)w)^2\\right)} y_i \\\\&= \\sum_{i=1}^n \\mathrm{softmax}\\left(-\\frac{1}{2}((x - x_i)w)^2\\right) y_i.\\end{aligned}$$\n",
|
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":eqlabel:`eq_nadaraya-watson-gaussian-para`\n",
|
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"\n",
|
||
"本节的余下部分将通过训练这个模型\n",
|
||
" :eqref:`eq_nadaraya-watson-gaussian-para`来学习注意力汇聚的参数。\n",
|
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"\n",
|
||
"### 批量矩阵乘法\n",
|
||
"\n",
|
||
":label:`subsec_batch_dot`\n",
|
||
"\n",
|
||
"为了更有效地计算小批量数据的注意力，\n",
|
||
"我们可以利用深度学习开发框架中提供的批量矩阵乘法。\n",
|
||
"\n",
|
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"假设第一个小批量数据包含$n$个矩阵$\\mathbf{X}_1,\\ldots, \\mathbf{X}_n$，\n",
|
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|
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"第二个小批量包含$n$个矩阵$\\mathbf{Y}_1, \\ldots, \\mathbf{Y}_n$，\n",
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|
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|
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|
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},
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"origin_pos": 33,
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"tab": [
|
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"pytorch"
|
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]
|
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},
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"outputs": [
|
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{
|
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"data": {
|
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"text/plain": [
|
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"torch.Size([2, 1, 6])"
|
||
]
|
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},
|
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"execution_count": 8,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
|
||
"X = torch.ones((2, 1, 4))\n",
|
||
"Y = torch.ones((2, 4, 6))\n",
|
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"torch.bmm(X, Y).shape"
|
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]
|
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},
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{
|
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"cell_type": "markdown",
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"id": "acac0804",
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"metadata": {
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"origin_pos": 36
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},
|
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"source": [
|
||
"在注意力机制的背景中，我们可以[**使用小批量矩阵乘法来计算小批量数据中的加权平均值**]。\n"
|
||
]
|
||
},
|
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{
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"execution_count": 9,
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"id": "7161cf35",
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},
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"origin_pos": 38,
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"tab": [
|
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"pytorch"
|
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]
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},
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"outputs": [
|
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{
|
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"data": {
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"text/plain": [
|
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"tensor([[[ 4.5000]],\n",
|
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"\n",
|
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" [[14.5000]]])"
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},
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"execution_count": 9,
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"metadata": {},
|
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"output_type": "execute_result"
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}
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],
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"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))"
|
||
]
|
||
},
|
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{
|
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"cell_type": "markdown",
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"id": "f31cb449",
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"metadata": {
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"origin_pos": 41
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},
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"source": [
|
||
"### 定义模型\n",
|
||
"\n",
|
||
"基于 :eqref:`eq_nadaraya-watson-gaussian-para`中的\n",
|
||
"[**带参数的注意力汇聚**]，使用小批量矩阵乘法，\n",
|
||
"定义Nadaraya-Watson核回归的带参数版本为：\n"
|
||
]
|
||
},
|
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{
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"cell_type": "code",
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"execution_count": 10,
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"id": "e7aee504",
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},
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"origin_pos": 43,
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"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"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(\n",
|
||
" -((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),\n",
|
||
" values.unsqueeze(-1)).reshape(-1)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
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"id": "192a922f",
|
||
"metadata": {
|
||
"origin_pos": 46
|
||
},
|
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"source": [
|
||
"### 训练\n",
|
||
"\n",
|
||
"接下来，[**将训练数据集变换为键和值**]用于训练注意力模型。\n",
|
||
"在带参数的注意力汇聚模型中，\n",
|
||
"任何一个训练样本的输入都会和除自己以外的所有训练样本的“键－值”对进行计算，\n",
|
||
"从而得到其对应的预测输出。\n"
|
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]
|
||
},
|
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{
|
||
"cell_type": "code",
|
||
"execution_count": 11,
|
||
"id": "c738e178",
|
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"metadata": {
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"execution": {
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"iopub.status.busy": "2023-08-18T07:07:17.362749Z",
|
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|
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"shell.execute_reply": "2023-08-18T07:07:17.368357Z"
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},
|
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"origin_pos": 48,
|
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"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [],
|
||
"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))"
|
||
]
|
||
},
|
||
{
|
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"cell_type": "markdown",
|
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"id": "245d8e27",
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"metadata": {
|
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"origin_pos": 51
|
||
},
|
||
"source": [
|
||
"[**训练带参数的注意力汇聚模型**]时，使用平方损失函数和随机梯度下降。\n"
|
||
]
|
||
},
|
||
{
|
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"cell_type": "code",
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"origin_pos": 53,
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"tab": [
|
||
"pytorch"
|
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]
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"source": [
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"如下所示，训练完带参数的注意力汇聚模型后可以发现：\n",
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"在尝试拟合带噪声的训练数据时，\n",
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"## 小结\n",
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"* Nadaraya-Watson核回归是具有注意力机制的机器学习范例。\n",
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"* 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",
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"## 练习\n",
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"\n",
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"1. 增加训练数据的样本数量，能否得到更好的非参数的Nadaraya-Watson核回归模型？\n",
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"1. 在带参数的注意力汇聚的实验中学习得到的参数$w$的价值是什么？为什么在可视化注意力权重时，它会使加权区域更加尖锐？\n",
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"1. 如何将超参数添加到非参数的Nadaraya-Watson核回归中以实现更好地预测结果？\n",
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||
"1. 为本节的核回归设计一个新的带参数的注意力汇聚模型。训练这个新模型并可视化其注意力权重。\n"
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