3.12.权重衰减.ipynb

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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 高维线性回归实验\n",
    "\n",
    "下面，我们以高维线性回归为例来引入一个过拟合问题，并使用权重衰减来应对过拟合。设数据样本特征的维度为$p$。对于训练数据集和测试数据集中特征为$x_1, x_2, \\ldots, x_p$的任一样本，我们使用如下的线性函数来生成该样本的标签：\n",
    "\n",
    "$$y = 0.05 + \\sum_{i = 1}^p 0.01x_i +  \\epsilon,$$\n",
    "\n",
    "其中噪声项$\\epsilon$服从均值为0、标准差为0.01的正态分布。为了较容易地观察过拟合，我们考虑高维线性回归问题，如设维度$p=200$；同时，我们特意把训练数据集的样本数设低，如20。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-05-14T08:04:33.582498Z",
     "start_time": "2019-05-14T08:04:32.887872Z"
    }
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import d2lzh as d2l\n",
    "from mxnet import autograd, gluon, init, nd\n",
    "from mxnet.gluon import data as gdata, loss as gloss, nn\n",
    "# n_train训练样本数，n_test测试样本数，num_inpyts输入特征数\n",
    "n_train, n_test, num_inputs = 20, 100, 200\n",
    "true_w, true_b = nd.ones((num_inputs, 1)) * 0.01, 0.05\n",
    "\n",
    "features = nd.random.normal(shape=(n_train + n_test, num_inputs))\n",
    "labels = nd.dot(features, true_w) + true_b\n",
    "labels += nd.random.normal(scale=0.01, shape=labels.shape)\n",
    "train_features, test_features = features[:n_train, :], features[n_train:, :]\n",
    "train_labels, test_labels = labels[:n_train], labels[n_train:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 从零开始实现"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 初始化模型参数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-05-14T08:09:14.776085Z",
     "start_time": "2019-05-14T08:09:14.772091Z"
    }
   },
   "outputs": [],
   "source": [
    "# 为每个参数附上梯度\n",
    "def init_params():\n",
    "    w = nd.random.normal(scale=1, shape=(num_inputs, 1))\n",
    "    b = nd.zeros(shape=(1,))\n",
    "    w.attach_grad()\n",
    "    b.attach_grad()\n",
    "    return [w, b]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 定义L_2范数惩罚项"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-05-14T08:14:29.077762Z",
     "start_time": "2019-05-14T08:14:29.074744Z"
    }
   },
   "outputs": [],
   "source": [
    "# 这里只惩罚模型的权重参数\n",
    "def l2_penalty(w):\n",
    "    return (w**2).sum() / 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 定义训练和测试"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-05-14T08:14:30.195615Z",
     "start_time": "2019-05-14T08:14:30.187634Z"
    }
   },
   "outputs": [],
   "source": [
    "batch_size, num_epochs, lr = 1, 100, 0.003\n",
    "# 线性模型和平方损失函数\n",
    "net, loss = d2l.linreg, d2l.squared_loss\n",
    "train_iter = gdata.DataLoader(gdata.ArrayDataset(\n",
    "    train_features, train_labels), batch_size, shuffle=True)\n",
    "\n",
    "def fit_and_plot(lambd):\n",
    "    w, b = init_params()\n",
    "    train_ls, test_ls = [], []\n",
    "    for _ in range(num_epochs):\n",
    "        for X, y in train_iter:\n",
    "            with autograd.record():\n",
    "                # 此处与之前不同， 添加了L2范数惩罚项\n",
    "                l = loss(net(X, w, b), y) + lambd * l2_penalty(w)\n",
    "            l.backward()\n",
    "            d2l.sgd([w, b], lr, batch_size)\n",
    "        train_ls.append(loss(net(train_features, w, b),\n",
    "                             train_labels).mean().asscalar())\n",
    "        test_ls.append(loss(net(test_features, w, b),\n",
    "                            test_labels).mean().asscalar())\n",
    "    d2l.semilogy(range(1, num_epochs + 1), train_ls, 'epochs', 'loss',\n",
    "                 range(1, num_epochs + 1), test_ls, ['train', 'test'])\n",
    "    print('L2 norm of w:', w.norm().asscalar())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 观察过拟合"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-05-14T08:14:34.720223Z",
     "start_time": "2019-05-14T08:14:31.602828Z"
    }
   },
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      "L2 norm of w: 12.904754\n"
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   ],
   "source": [
    "# lambd为0，没有用权重衰减，典型的过拟合\n",
    "fit_and_plot(lambd=0)"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 使用权重衰减"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-05-14T08:15:39.899115Z",
     "start_time": "2019-05-14T08:15:36.891474Z"
    }
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     "metadata": {
      "needs_background": "light"
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     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "L2 norm of w: 0.042179428\n"
     ]
    }
   ],
   "source": [
    "# 虽然训练误差比刚才大，但是测试集上的误差有所下降，一定程度的缓解了过拟合，\n",
    "# 另外，权重参数的L_2范数比不使用权重衰减时的更小，此时的权重参数更接近0\n",
    "fit_and_plot(lambd=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 简洁实现"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-05-14T08:18:21.196737Z",
     "start_time": "2019-05-14T08:18:21.187761Z"
    }
   },
   "outputs": [],
   "source": [
    "def fit_and_plot_gluon(wd):\n",
    "    net = nn.Sequential()\n",
    "    net.add(nn.Dense(1))\n",
    "    net.initialize(init.Normal(sigma=1))\n",
    "    # 对权重参数衰减。权重名称一般是以weight结尾，加参数wd，wd就是前面的lambda值\n",
    "    trainer_w = gluon.Trainer(net.collect_params('.*weight'), 'sgd',\n",
    "                              {'learning_rate': lr, 'wd': wd})\n",
    "    # 不对偏差参数衰减。偏差名称一般是以bias结尾\n",
    "    trainer_b = gluon.Trainer(net.collect_params('.*bias'), 'sgd',\n",
    "                              {'learning_rate': lr})\n",
    "    train_ls, test_ls = [], []\n",
    "    for _ in range(num_epochs):\n",
    "        for X, y in train_iter:\n",
    "            with autograd.record():\n",
    "                l = loss(net(X), y)\n",
    "            l.backward()\n",
    "            # 对两个Trainer实例分别调用step函数，从而分别更新权重和偏差\n",
    "            trainer_w.step(batch_size)\n",
    "            trainer_b.step(batch_size)\n",
    "        train_ls.append(loss(net(train_features),\n",
    "                             train_labels).mean().asscalar())\n",
    "        test_ls.append(loss(net(test_features),\n",
    "                            test_labels).mean().asscalar())\n",
    "    d2l.semilogy(range(1, num_epochs + 1), train_ls, 'epochs', 'loss',\n",
    "                 range(1, num_epochs + 1), test_ls, ['train', 'test'])\n",
    "    print('L2 norm of w:', net[0].weight.data().norm().asscalar())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-05-14T08:18:30.296884Z",
     "start_time": "2019-05-14T08:18:28.371007Z"
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    "## 小结\n",
    "\n",
    "* 正则化通过为模型损失函数添加惩罚项使学出的模型参数值较小，是应对过拟合的常用手段。\n",
    "* 权重衰减等价于$L_2$范数正则化，通常会使学到的权重参数的元素较接近0。\n",
    "* 权重衰减可以通过Gluon的`wd`超参数来指定。\n",
    "* 可以定义多个`Trainer`实例对不同的模型参数使用不同的迭代方法。\n",
    "\n",
    "## 练习\n",
    "\n",
    "* 回顾一下训练误差和泛化误差的关系。除了权重衰减、增大训练量以及使用复杂度合适的模型，你还能想到哪些办法来应对过拟合？\n",
    "* 如果你了解贝叶斯统计，你觉得权重衰减对应贝叶斯统计里的哪个重要概念？\n",
    "* 调节实验中的权重衰减超参数，观察并分析实验结果。"
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