code/12-权重衰退.ipynb

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   "execution_count": 1,
   "id": "8d522eed",
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import torch\n",
    "from torch import nn\n",
    "from d2l import torch as d2l"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "41c43196",
   "metadata": {},
   "source": [
    "生成数据公式\n",
    "$$y = 0.05 + \\sum_{i = 1}^d 0.01 x_i + \\epsilon \\text{ where }\n",
    "\\epsilon \\sim \\mathcal{N}(0, 0.01^2).$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "b29aaed5",
   "metadata": {},
   "outputs": [],
   "source": [
    "#制作人工数据集\n",
    "n_train, n_test, num_inputs, batch_size = 20, 100, 200, 5 #特意将训练集做的比较小，使得容易发生过拟合\n",
    "true_w, true_b = torch.ones((num_inputs, 1)) * 0.01, 0.05 #w真实值全为0.01，b为0.05\n",
    "train_data = d2l.synthetic_data(true_w, true_b, n_test) #生成数据集\n",
    "train_iter = d2l.load_array(train_data, batch_size) #从数组生成iterate\n",
    "test_data = d2l.synthetic_data(true_w, true_b, n_test)\n",
    "test_iter = d2l.load_array(test_data, batch_size, is_train=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "ee8e1d4b",
   "metadata": {},
   "outputs": [],
   "source": [
    "#初始化模型参数\n",
    "def init_params():\n",
    "    w = torch.normal(0, 1, size=(num_inputs, 1),requires_grad=True)#均值为0，方差为1\n",
    "    b = torch.zeros(1, requires_grad=True)\n",
    "    return [w, b]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e4f6e3da",
   "metadata": {},
   "source": [
    "定义惩罚，这里改为使用L1范数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "86577c80",
   "metadata": {},
   "outputs": [],
   "source": [
    "def l1_penalty(w):\n",
    "    return torch.sum(torch.abs(w))#对所有元素取绝对值并求和"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f0b39ef6",
   "metadata": {},
   "source": [
    "训练函数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "156a538a",
   "metadata": {},
   "outputs": [],
   "source": [
    "def train(lambd):#接受超参数lambda\n",
    "    w, b = init_params()#初始化参数\n",
    "    net, loss = lambda X: d2l.linreg(X, w, b), d2l.squared_loss #线性回归，平方损失，lambda为匿名函数\n",
    "    num_epochs, lr = 100, 0.003 #迭代数与学习率\n",
    "    animator = d2l.Animator(xlabel='epochs', ylabel='loss', yscale='log',\n",
    "                           xlim=[5,num_epochs], legend=['train', 'test'])#作图\n",
    "    for epoch in range(num_epochs):#训练迭代\n",
    "        for X, y in train_iter:\n",
    "            l = loss(net(X), y)+ lambd * l1_penalty(w)#计算损失，即原损失+正则项\n",
    "            #严格地讲使用L1范数时表达式的形式不应该是这样，见练习3\n",
    "            l.sum().backward()#对一个batch的损失求和后反向传播\n",
    "            d2l.sgd([w, b], lr, batch_size)\n",
    "        if (epoch + 1) % 5 == 0:#每5迭代打印一次\n",
    "            animator.add(epoch + 1, (d2l.evaluate_loss(net, train_iter, loss),\n",
    "                                    d2l.evaluate_loss(net, test_iter, loss)))\n",
    "    print('w的L1范数是：', torch.sum(torch.abs(w)).item())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "5cfd6690",
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   "outputs": [
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     "name": "stdout",
     "output_type": "stream",
     "text": [
      "w的L1范数是： 116.36851501464844\n"
     ]
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   "source": [
    "使用权重衰减\n",
    "可以自己多换几个$\\lambda$取值看看不同效果"
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      "w的L1范数是： 0.8542037606239319\n"
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   "source": [
    "train(lambd=3)"
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  {
   "cell_type": "markdown",
   "id": "37a7f445",
   "metadata": {},
   "source": [
    "简洁实现，直接使用weight_decay指定超参数，是用pytorch实现的，所以这里是L2范数。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "5ee95030",
   "metadata": {},
   "outputs": [],
   "source": [
    "def train_concise(wd):\n",
    "    net = nn.Sequential(nn.Linear(num_inputs, 1))\n",
    "    for param in net.parameters():\n",
    "        param.data.normal_()\n",
    "    loss = nn.MSELoss(reduction='none')\n",
    "    num_epochs, lr = 100, 0.003\n",
    "    # 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.sum().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())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "9c1384c8",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "w的L2范数： 8.456375122070312\n"
     ]
    },
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      "w的L2范数： 0.09245594590902328\n"
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    "#### 练习（答案仅供参考）\n",
    "1. 在本节的估计问题中使用$\\lambda$的值进行实验。绘制训练和测试精度关于$\\lambda$的函数。你观察到了什么？\n",
    "<p>随lambda增大精度先降低后不收敛，"
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   "source": [
    "[评论区](#https://zh-v2.d2l.ai/chapter_multilayer-perceptrons/weight-decay.html)有其他同学发布的图片"
   ]
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    "2. 使用验证集来找到最佳值$\\lambda$。它真的是最优值吗？这有关系吗？\n",
    "<p>不能确定是否是最优值，但没关系，只要足够优即可\n",
    "1. 如果我们使用$\\sum_i |w_i|$作为我们选择的惩罚（$L_1$正则化），那么更新方程会是什么样子？\n",
    "<p>$\\frac{\\partial{}}{\\partial{\\mathbf{w}}}(\\ell(\\mathbf{w},b)+\\lambda||\\mathbf{w}||_1)=\\frac{\\partial{\\ell(\\mathbf{w},b)}}{{\\partial{\\mathbf{w}}}}+\\lambda I'$\n",
    "<p>其中$I'=(a_1,...,a_n)$,当$\\mathbf{w}$中第$i$个元素为正时$a_i=1$，反之$a_i=-1$.（=0时随意）<p>代入公式化简得$\\mathbf{w}_{t+1}=\\mathbf{w}_{t}-\\eta\\frac{\\partial{\\ell(\\mathbf{w}_t,b_t)}}{{\\partial{\\mathbf{w}_{t}}}}-\\eta\\lambda I'$\n",
    "<p>从这个式子可以看出使用L1正则化时只能对所有同号的参数施加一个相同大小的正则项（增减一个定值），而反观L2正则化对参数的影响是与参数本身的值有关的（乘上一个系数）似乎是更好的选择。不过L1正则化在特征提取上会有用处。\n",
    "1. 我们知道$\\|\\mathbf{w}\\|^2 = \\mathbf{w}^\\top \\mathbf{w}$。你能找到类似的矩阵方程吗（见 :numref:`subsec_lin-algebra-norms` 中的Frobenius范数）？<p>这个不太理解题意，可能是说对称矩阵？\n",
    "1. 回顾训练误差和泛化误差之间的关系。除了权重衰减、增加训练数据、使用适当复杂度的模型之外，你还能想出其他什么方法来处理过拟合？<p>EarlyStopping等\n",
    "1. 在贝叶斯统计中，我们使用先验和似然的乘积，通过公式$P(w \\mid x) \\propto P(x \\mid w) P(w)$得到后验。如何得到带正则化的$P(w)$？\n",
    "\n"
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