9187 lines
339 KiB
Plaintext
9187 lines
339 KiB
Plaintext
{
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"cells": [
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"cell_type": "markdown",
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"id": "6b83a647-6801-4667-a822-0656b8e9556a",
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"metadata": {},
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"source": [
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"# 模型选择\n",
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"\n",
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"目录\n",
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"\n",
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"- 欠拟合和过拟合\n",
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"- 权重衰减\n",
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"- 暂退法（Dropout）"
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]
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},
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"cell_type": "markdown",
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"id": "30394f0b-c857-48e6-b3f5-b7ceba0411c8",
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"slide_type": "-"
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},
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"tags": []
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"source": [
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"## 欠拟合和过拟合\n",
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"\n",
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"通过多项式拟合来探索这些概念"
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]
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"cell_type": "code",
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"execution_count": 1,
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"id": "70d264d4-0dae-48f1-9288-375f9d84aa10",
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"metadata": {
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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 math\n",
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"import numpy as np\n",
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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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"id": "68ca9f7a-b179-4fd0-a183-65ce654526d7",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"使用以下三阶多项式来生成训练和测试数据的标签：\n",
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"$$y = 5 + 1.2x - 3.4\\frac{x^2}{2!} + 5.6 \\frac{x^3}{3!} + \\epsilon \\text{ where }\n",
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"\\epsilon \\sim \\mathcal{N}(0, 0.1^2)$$"
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]
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"cell_type": "code",
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"execution_count": 2,
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"id": "513f2378-8df6-4eaa-b98a-59c2b890fbae",
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"metadata": {
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"origin_pos": 5,
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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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"max_degree = 20 # 多项式的最大阶数\n",
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"n_train, n_test = 100, 100 # 训练和测试数据集大小\n",
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"true_w = np.zeros(max_degree) # 分配大量的空间\n",
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"true_w[0:4] = np.array([5, 1.2, -3.4, 5.6])\n",
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"\n",
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"features = np.random.normal(size=(n_train + n_test, 1))\n",
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"np.random.shuffle(features)\n",
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"poly_features = np.power(features, np.arange(max_degree).reshape(1, -1))\n",
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"for i in range(max_degree):\n",
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" poly_features[:, i] /= math.gamma(i + 1) # gamma(n)=(n-1)!\n",
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"# labels的维度:(n_train+n_test,)\n",
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"labels = np.dot(poly_features, true_w)\n",
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"labels += np.random.normal(scale=0.1, size=labels.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": "eda1d879-8d72-40bb-9d01-8aaa5ca28f4e",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"查看一下前2个样本"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "a1988415-3bc9-4e2b-bb2c-8f10d26b97e1",
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"metadata": {
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"origin_pos": 8,
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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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"data": {
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"(tensor([[0.7135],\n",
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" [0.4976]]),\n",
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" tensor([[1.0000e+00, 7.1346e-01, 2.5451e-01, 6.0527e-02, 1.0796e-02, 1.5405e-03,\n",
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" 1.8318e-04, 1.8670e-05, 1.6650e-06, 1.3199e-07, 9.4170e-09, 6.1079e-10,\n",
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" 3.6314e-11, 1.9930e-12, 1.0156e-13, 4.8308e-15, 2.1541e-16, 9.0403e-18,\n",
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" 3.5833e-19, 1.3455e-20],\n",
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" [1.0000e+00, 4.9764e-01, 1.2382e-01, 2.0540e-02, 2.5554e-03, 2.5434e-04,\n",
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" 2.1095e-05, 1.4997e-06, 9.3288e-08, 5.1582e-09, 2.5670e-10, 1.1613e-11,\n",
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" 4.8160e-13, 1.8436e-14, 6.5531e-16, 2.1741e-17, 6.7620e-19, 1.9794e-20,\n",
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" 5.4725e-22, 1.4334e-23]]),\n",
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" tensor([5.3958, 5.5251]))"
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]
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},
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"execution_count": 3,
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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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"# NumPy ndarray转换为tensor\n",
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"true_w, features, poly_features, labels = [torch.tensor(x, dtype=\n",
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" torch.float32) for x in [true_w, features, poly_features, labels]]\n",
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"\n",
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"features[:2], poly_features[:2, :], labels[:2]"
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]
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},
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{
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"cell_type": "markdown",
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"id": "f39a6b66-4877-415f-88e7-fee01a15283f",
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"实现一个函数来评估模型在给定数据集上的损失"
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]
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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": "a3a49a16-7428-46f9-89e9-44ae554a182d",
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"metadata": {
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"origin_pos": 11,
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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": [
|
||
"def evaluate_loss(net, data_iter, loss): \n",
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" \"\"\"评估给定数据集上模型的损失\"\"\"\n",
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||
" metric = d2l.Accumulator(2) # 损失的总和,样本数量\n",
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||
" for X, y in data_iter:\n",
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" out = net(X)\n",
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||
" y = y.reshape(out.shape)\n",
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||
" l = loss(out, y)\n",
|
||
" metric.add(l.sum(), l.numel())\n",
|
||
" return metric[0] / metric[1]"
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]
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},
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{
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"cell_type": "markdown",
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"id": "1d3ab43b-025e-47fa-a787-b8e051703282",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"定义训练函数"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"id": "3b3a4d31-223a-43ba-9e05-751078977a7c",
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"metadata": {
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"origin_pos": 14,
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"tab": [
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"pytorch"
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]
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"source": [
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"def train(train_features, test_features, train_labels, test_labels,\n",
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" num_epochs=400):\n",
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" loss = nn.MSELoss(reduction='none')\n",
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||
" input_shape = train_features.shape[-1]\n",
|
||
" # 不设置偏置，因为我们已经在多项式中实现了它\n",
|
||
" net = nn.Sequential(nn.Linear(input_shape, 1, bias=False))\n",
|
||
" batch_size = min(10, train_labels.shape[0])\n",
|
||
" train_iter = d2l.load_array((train_features, train_labels.reshape(-1,1)),\n",
|
||
" batch_size)\n",
|
||
" test_iter = d2l.load_array((test_features, test_labels.reshape(-1,1)),\n",
|
||
" batch_size, is_train=False)\n",
|
||
" trainer = torch.optim.SGD(net.parameters(), lr=0.01)\n",
|
||
" animator = d2l.Animator(xlabel='epoch', ylabel='loss', yscale='log',\n",
|
||
" xlim=[1, num_epochs], ylim=[1e-3, 1e2],\n",
|
||
" legend=['train', 'test'])\n",
|
||
" for epoch in range(num_epochs):\n",
|
||
" d2l.train_epoch_ch3(net, train_iter, loss, trainer)\n",
|
||
" if epoch == 0 or (epoch + 1) % 20 == 0:\n",
|
||
" animator.add(epoch + 1, (evaluate_loss(net, train_iter, loss),\n",
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||
" evaluate_loss(net, test_iter, loss)))\n",
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" print('weight:', net[0].weight.data.numpy())"
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]
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},
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{
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"cell_type": "markdown",
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"slide_type": "slide"
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},
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"source": [
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"三阶多项式函数拟合(正常)"
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]
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"# 从多项式特征中选取所有维度\n",
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"%matplotlib inline\n",
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]
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}
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"像以前一样生成一些数据\n",
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"metadata": {
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"origin_pos": 5,
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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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"n_train, n_test, num_inputs, batch_size = 20, 100, 200, 5\n",
|
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"true_w, true_b = torch.ones((num_inputs, 1)) * 0.01, 0.05\n",
|
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"train_data = d2l.synthetic_data(true_w, true_b, n_train)\n",
|
||
"train_iter = d2l.load_array(train_data, batch_size)\n",
|
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"test_data = d2l.synthetic_data(true_w, true_b, n_test)\n",
|
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"test_iter = d2l.load_array(test_data, batch_size, is_train=False)"
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]
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},
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{
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
|
||
"初始化模型参数。注意：$w$的标准差异常高，容易导致优化问题"
|
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]
|
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"id": "f38cd2b7-2537-4c6c-9157-804cdde17ff9",
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"metadata": {
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||
"origin_pos": 8,
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"tab": [
|
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"pytorch"
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||
]
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||
},
|
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"outputs": [],
|
||
"source": [
|
||
"def init_params():\n",
|
||
" w = torch.normal(0, 1, size=(num_inputs, 1), requires_grad=True)\n",
|
||
" b = torch.zeros(1, requires_grad=True)\n",
|
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" return [w, b]"
|
||
]
|
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},
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{
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"cell_type": "markdown",
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"id": "e5875ed8-6ae1-46ca-82fc-7b0eaf870aeb",
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"metadata": {
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"slideshow": {
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"slide_type": "-"
|
||
}
|
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},
|
||
"source": [
|
||
"定义$L_2$范数惩罚"
|
||
]
|
||
},
|
||
{
|
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"cell_type": "code",
|
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"id": "9e70a101-c049-481f-888c-76c492432cc1",
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"metadata": {
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"origin_pos": 12,
|
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"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
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"outputs": [],
|
||
"source": [
|
||
"def l2_penalty(w):\n",
|
||
" return torch.sum(w.pow(2)) / 2"
|
||
]
|
||
},
|
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{
|
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"id": "1dcb1126-4b76-416c-ac79-4813f759f890",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
|
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},
|
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"source": [
|
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"定义训练代码实现"
|
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]
|
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},
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{
|
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"cell_type": "code",
|
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"execution_count": 13,
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"id": "970f04ea-eaab-436c-8044-6cc01390e530",
|
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"metadata": {
|
||
"origin_pos": 16,
|
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"tab": [
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|
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]
|
||
},
|
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"outputs": [],
|
||
"source": [
|
||
"def train(lambd):\n",
|
||
" w, b = init_params()\n",
|
||
" net, loss = lambda X: d2l.linreg(X, w, b), d2l.squared_loss\n",
|
||
" num_epochs, lr = 100, 0.003\n",
|
||
" animator = d2l.Animator(xlabel='epochs', ylabel='loss', yscale='log',\n",
|
||
" xlim=[5, num_epochs], legend=['train', 'test'])\n",
|
||
" for epoch in range(num_epochs):\n",
|
||
" for X, y in train_iter:\n",
|
||
" # 广播机制使l2_penalty(w)成为一个长度为batch_size的向量\n",
|
||
" l = loss(net(X), y) + lambd * l2_penalty(w) # 增加了L2范数惩罚项\n",
|
||
" l.sum().backward()\n",
|
||
" d2l.sgd([w, b], lr, batch_size)\n",
|
||
" if (epoch + 1) % 5 == 0:\n",
|
||
" animator.add(epoch + 1, (d2l.evaluate_loss(net, train_iter, loss),\n",
|
||
" d2l.evaluate_loss(net, test_iter, loss)))\n",
|
||
" print('w的L2范数是：', torch.norm(w).item())"
|
||
]
|
||
},
|
||
{
|
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"cell_type": "markdown",
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"id": "14e879f7-bee3-49bf-acc8-0b33510de08f",
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"slideshow": {
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"slide_type": "slide"
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}
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},
|
||
"source": [
|
||
"忽略正则化直接训练。非常典型的欠拟合：训练误差还在下降阶段"
|
||
]
|
||
},
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{
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"id": "7670e9cc-28ea-4dc5-ad88-53189f3ce1b3",
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"metadata": {
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"origin_pos": 19,
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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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"name": "stdout",
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"output_type": "stream",
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"text": [
|
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"w的L2范数是： 13.431254386901855\n"
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]
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" # 偏置参数没有衰减\n",
|
||
" trainer = torch.optim.SGD([\n",
|
||
" {\"params\":net[0].weight,'weight_decay': wd},\n",
|
||
" {\"params\":net[0].bias}], lr=lr)\n",
|
||
" animator = d2l.Animator(xlabel='epochs', ylabel='loss', yscale='log',\n",
|
||
" xlim=[5, num_epochs], legend=['train', 'test'])\n",
|
||
" for epoch in range(num_epochs):\n",
|
||
" for X, y in train_iter:\n",
|
||
" trainer.zero_grad()\n",
|
||
" l = loss(net(X), y)\n",
|
||
" l.mean().backward()\n",
|
||
" trainer.step()\n",
|
||
" if (epoch + 1) % 5 == 0:\n",
|
||
" animator.add(epoch + 1,\n",
|
||
" (d2l.evaluate_loss(net, train_iter, loss),\n",
|
||
" d2l.evaluate_loss(net, test_iter, loss)))\n",
|
||
" print('w的L2范数：', net[0].weight.norm().item())"
|
||
]
|
||
},
|
||
{
|
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"cell_type": "markdown",
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"id": "081cf988-6c1b-46fe-8af7-986b387581dc",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
|
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},
|
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"source": [
|
||
"这些图看起来与之前类似。但训练误差更快趋向水平：库函数有内部优化"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
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"execution_count": 17,
|
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"id": "809320c9-7ef9-414c-9df2-a10b36f6e8ce",
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"metadata": {
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"origin_pos": 30,
|
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"tab": [
|
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"pytorch"
|
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]
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||
},
|
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"outputs": [
|
||
{
|
||
"name": "stdout",
|
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"output_type": "stream",
|
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"text": [
|
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"w的L2范数： 14.588664054870605\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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"slide_type": "slide"
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}
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},
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"source": [
|
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"测试`dropout_layer`函数"
|
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},
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{
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
|
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"tensor([[ 0., 1., 2., 3., 4., 5., 6., 7.],\n",
|
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" [ 8., 9., 10., 11., 12., 13., 14., 15.]])\n",
|
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"tensor([[ 0., 1., 2., 3., 4., 5., 6., 7.],\n",
|
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" [ 8., 9., 10., 11., 12., 13., 14., 15.]])\n",
|
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|
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"tensor([[0., 0., 0., 0., 0., 0., 0., 0.],\n",
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" [0., 0., 0., 0., 0., 0., 0., 0.]])\n"
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]
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}
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],
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"source": [
|
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"X= torch.arange(16, dtype = torch.float32).reshape((2, 8))\n",
|
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"print(X)\n",
|
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"print(dropout_layer(X, 0.))\n",
|
||
"print(dropout_layer(X, 0.5))\n",
|
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"print(dropout_layer(X, 1.))"
|
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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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"slideshow": {
|
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"slide_type": "slide"
|
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}
|
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},
|
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"source": [
|
||
"定义具有两个隐藏层的多层感知机，每个隐藏层包含256个单元"
|
||
]
|
||
},
|
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{
|
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"cell_type": "code",
|
||
"execution_count": 21,
|
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"id": "26c4ba3a-48e1-4eac-86da-c4846e0ac04e",
|
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"metadata": {
|
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"origin_pos": 14,
|
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"tab": [
|
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"pytorch"
|
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]
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"num_inputs, num_outputs, num_hiddens1, num_hiddens2 = 784, 10, 256, 256\n",
|
||
"\n",
|
||
"dropout1, dropout2 = 0.2, 0.5\n",
|
||
"\n",
|
||
"class Net(nn.Module):\n",
|
||
" def __init__(self, num_inputs, num_outputs, num_hiddens1, num_hiddens2,\n",
|
||
" is_training = True):\n",
|
||
" super(Net, self).__init__()\n",
|
||
" self.num_inputs = num_inputs\n",
|
||
" self.training = is_training\n",
|
||
" self.lin1 = nn.Linear(num_inputs, num_hiddens1)\n",
|
||
" self.lin2 = nn.Linear(num_hiddens1, num_hiddens2)\n",
|
||
" self.lin3 = nn.Linear(num_hiddens2, num_outputs)\n",
|
||
" self.relu = nn.ReLU()\n",
|
||
"\n",
|
||
" def forward(self, X):\n",
|
||
" H1 = self.relu(self.lin1(X.reshape((-1, self.num_inputs))))\n",
|
||
" if self.training == True: # 只有在训练模型时才使用dropout\n",
|
||
" H1 = dropout_layer(H1, dropout1) # 在第一个全连接层之后添加一个dropout层\n",
|
||
" H2 = self.relu(self.lin2(H1))\n",
|
||
" if self.training == True:\n",
|
||
" H2 = dropout_layer(H2, dropout2) # 在第一个全连接层之后添加一个dropout层\n",
|
||
" out = self.lin3(H2)\n",
|
||
" return out\n",
|
||
"\n",
|
||
"\n",
|
||
"net = Net(num_inputs, num_outputs, num_hiddens1, num_hiddens2)"
|
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]
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},
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{
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"slideshow": {
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"slide_type": "slide"
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}
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},
|
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"source": [
|
||
"训练和测试"
|
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]
|
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},
|
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{
|
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"cell_type": "code",
|
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"execution_count": 22,
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"origin_pos": 18,
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"tab": [
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"pytorch"
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]
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"net = nn.Sequential(nn.Flatten(),\n",
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" nn.Linear(784, 256),\n",
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" nn.ReLU(),\n",
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" nn.Dropout(dropout1), # 在第一个全连接层之后添加一个dropout层\n",
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" nn.Linear(256, 256),\n",
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