3.2.线性回归的从零开始实现.ipynb

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   "source": [
    "%matplotlib inline\n",
    "from IPython import display\n",
    "from matplotlib import pyplot as plt\n",
    "from mxnet import autograd, nd\n",
    "import random"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 生成数据集"
   ]
  },
  {
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   "source": [
    "# 根据自己设定的权重和偏置加上一些噪音来生成一些数据，用这些数据去拟合自己设定的权重和偏置\n",
    "# 输入样本的特征数\n",
    "num_inputs = 2\n",
    "# 输入样本的个数\n",
    "num_examples = 1000\n",
    "# 真实的权重和偏置\n",
    "true_w = [2, -3.4]\n",
    "true_b = 4.2\n",
    "# 随机生成均值0标准差1的正态分布数据，形状是（样本数，特征数）\n",
    "features = nd.random.normal(scale=1, shape=(num_examples, num_inputs))\n",
    "# 根据𝑦=𝑋𝑤+𝑏，来生成结果y，形状是（样本数，1），这里加true_b涉及广播\n",
    "labels = true_w[0] * features[:, 0] + true_w[1] * features[:, 1] + true_b\n",
    "# 给这些y加上一些噪音，不如拟合出来一定是固定的true_w和true_b\n",
    "# 噪音服从均值为0、标准差为0.01的正态分布，形状和输出y的形状一样\n",
    "labels += nd.random.normal(scale=0.01, shape=labels.shape)"
   ]
  },
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   "source": [
    "# features的每一行是一个长度为2的向量，而labels的每一行是一个长度为1的向量（标量）。\n",
    "features[0], labels[0]"
   ]
  },
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      "text/plain": [
       "<Figure size 252x180 with 1 Axes>"
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     },
     "metadata": {
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     "output_type": "display_data"
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   ],
   "source": [
    "# 此处画图无个人注释\n",
    "# 通过生成第二个特征features[:, 1]和标签 labels 的散点图，\n",
    "# 可以更直观地观察两者间的线性关系。\n",
    "def use_svg_display():\n",
    "    # 用矢量图显示\n",
    "    display.set_matplotlib_formats('svg')\n",
    "\n",
    "def set_figsize(figsize=(3.5, 2.5)):\n",
    "    use_svg_display()\n",
    "    # 设置图的尺寸\n",
    "    plt.rcParams['figure.figsize'] = figsize\n",
    "\n",
    "set_figsize()\n",
    "plt.scatter(features[:, 1].asnumpy(), labels.asnumpy(), 1);  # 加分号只显示图"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 读取数据"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-05-09T09:13:49.923491Z",
     "start_time": "2019-05-09T09:13:49.917509Z"
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   "outputs": [],
   "source": [
    "# 生成minibatch的函数\n",
    "# 输入批量大小，样本和标签，返回num_samples/batch_size 个批量数据(假设整除)\n",
    "def data_iter(batch_size, features, labels):\n",
    "    num_examples = len(features)# 样本个数\n",
    "    index = list(range(num_examples))# 为每个样本来个index\n",
    "    random.shuffle(index) # 随机打乱index，index对应样本，也就打乱了样本\n",
    "    # 0到num_examples之间，以batch_size为步长取数，如0-10，3为步长，取出的是0369\n",
    "    for i in range(0, num_examples, batch_size):\n",
    "        # j是以i为开始，取到i+batch_size这一段，\n",
    "        # 或者i到num_examples这一段（当i+batch_size > num_examples了），\n",
    "        # 注意，这一段的内容是样本的index，并且前面打乱了\n",
    "        j = nd.array(index[i: min(i + batch_size, num_examples)])\n",
    "        # 从样本和标签中把取到的这一段index数据所对应的样本和标签值提取出来，并返回出来\n",
    "        yield features.take(j), labels.take(j)"
   ]
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   "execution_count": 6,
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     "start_time": "2019-05-09T09:13:49.924490Z"
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    "scrolled": true
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "[[-0.12857106 -0.20730972]\n",
      " [-0.25380394 -0.2829764 ]\n",
      " [ 0.34802237  0.2528925 ]\n",
      " [-0.6517895  -0.56789935]\n",
      " [ 0.88323224  1.104073  ]\n",
      " [ 1.5444453   0.7583689 ]\n",
      " [ 1.7005018   0.25498316]\n",
      " [ 0.29402134 -1.5258901 ]\n",
      " [ 0.69407785 -1.6696625 ]\n",
      " [-1.2315758  -1.9084588 ]]\n",
      "<NDArray 10x2 @cpu(0)> \n",
      "[ 4.670208   4.658644   4.0320587  4.81403    2.2122235  4.689574\n",
      "  6.736409   9.982809  11.26492    8.250098 ]\n",
      "<NDArray 10 @cpu(0)>\n",
      "\n",
      "[[-0.6464501   0.8603266 ]\n",
      " [ 0.9916597   2.0971553 ]\n",
      " [ 0.29956347  0.15302546]\n",
      " [-0.26981238 -0.5396458 ]\n",
      " [ 1.6553524   2.009465  ]\n",
      " [-0.21794528  0.26581082]\n",
      " [ 1.8307649  -1.1468065 ]\n",
      " [ 2.2202117   1.7947233 ]\n",
      " [ 0.09339606  0.8894987 ]\n",
      " [-0.507296   -2.6159513 ]]\n",
      "<NDArray 10x2 @cpu(0)> \n",
      "[-0.03009245 -0.94586164  4.2968144   5.4882474   0.6711063   2.84934\n",
      " 11.751628    2.5394602   1.3711476  12.062984  ]\n",
      "<NDArray 10 @cpu(0)>\n",
      "\n",
      "[[-0.8204806   0.34175402]\n",
      " [ 1.2824047  -0.54583657]\n",
      " [ 2.0454583   0.570716  ]\n",
      " [ 1.7353765  -1.0301763 ]\n",
      " [-1.3868748  -0.7748142 ]\n",
      " [-0.76957184 -0.5510532 ]\n",
      " [ 0.27856717 -2.6699395 ]\n",
      " [-2.513012    1.205448  ]\n",
      " [-0.6570352  -0.7010761 ]\n",
      " [-0.12680836  1.2730464 ]]\n",
      "<NDArray 10x2 @cpu(0)> \n",
      "[ 1.4108375   8.617071    6.3560915  11.178011    4.0478897   4.542024\n",
      " 13.8299885  -4.9235134   5.275559   -0.37338325]\n",
      "<NDArray 10 @cpu(0)>\n"
     ]
    }
   ],
   "source": [
    "# 批量大小是10，样本数1000\n",
    "batch_size = 10\n",
    "\n",
    "n = 0\n",
    "for X, y in data_iter(batch_size, features, labels):\n",
    "    n += 1\n",
    "    print(X, y)\n",
    "    if n >2:\n",
    "        break"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-05-09T07:48:32.252337Z",
     "start_time": "2019-05-09T07:48:32.248350Z"
    }
   },
   "source": [
    "## 初始化模型参数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-05-09T09:13:49.956404Z",
     "start_time": "2019-05-09T09:13:49.949423Z"
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   },
   "outputs": [],
   "source": [
    "# 我们将权重初始化成均值为0、标准差为0.01的正态随机数，形状是（特征数，1）\n",
    "# 偏差则初始化成0\n",
    "w = nd.random.normal(scale=0.01, shape=(num_inputs, 1))\n",
    "b = nd.zeros(shape=(1,))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-05-09T09:13:49.969370Z",
     "start_time": "2019-05-09T09:13:49.959396Z"
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   },
   "outputs": [],
   "source": [
    "# 创建它们的梯度\n",
    "w.attach_grad()\n",
    "b.attach_grad()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 定义模型"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-05-09T09:13:49.981337Z",
     "start_time": "2019-05-09T09:13:49.973359Z"
    }
   },
   "outputs": [],
   "source": [
    "def linreg(X, w, b):  # 本函数已保存在d2lzh包中方便以后使用\n",
    "    return nd.dot(X, w) + b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 定义损失函数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-05-09T09:13:49.994303Z",
     "start_time": "2019-05-09T09:13:49.983332Z"
    }
   },
   "outputs": [],
   "source": [
    "# 平方损失函数，需要把真实值y变形成预测值y_hat的形状\n",
    "# 输入预测值，与真实值，输出是一个向量\n",
    "def squared_loss(y_hat, y):  # 本函数已保存在d2lzh包中方便以后使用\n",
    "    return (y_hat - y.reshape(y_hat.shape)) ** 2 / 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 定义优化算法"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-05-09T09:13:50.007268Z",
     "start_time": "2019-05-09T09:13:49.996298Z"
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   },
   "outputs": [],
   "source": [
    "# 小批量随机梯度下降，这里的小批量随机其实体现在data_iter中，而sgd只是求梯度更新参数\n",
    "def sgd(params, lr, batch_size):  # 本函数已保存在d2lzh包中方便以后使用\n",
    "    # 对每一个参数，这里每一个不是指每一个w,传入的是[w, b]，那么就是对w更新，对b更新\n",
    "    # w是向量，同时对所有权重更新\n",
    "    for param in params:\n",
    "        #这里自动求梯度模块计算得来的梯度是一个批量样本的梯度和\n",
    "        #所以我们将它除以批量大小来得到平均值。\n",
    "        param[:] = param - lr * param.grad / batch_size"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 训练模型"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-05-09T09:13:50.373293Z",
     "start_time": "2019-05-09T09:13:50.009263Z"
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   },
   "outputs": [
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     "text": [
      "epoch 1, loss 0.034961\n",
      "epoch 2, loss 0.000124\n",
      "epoch 3, loss 0.000049\n"
     ]
    }
   ],
   "source": [
    "lr = 0.03 # 学习率（步长）\n",
    "num_epochs = 3 # 要训练的周期数，轮数\n",
    "net = linreg # 模型是线性回归\n",
    "loss = squared_loss # 损失函数是平方损失\n",
    "\n",
    "for epoch in range(num_epochs):  # 训练模型一共需要num_epochs个迭代周期\n",
    "    # 在每一个迭代周期中，会使用训练数据集中所有样本一次（假设样本数能够被批量大小整除）。X\n",
    "    # X和y分别是小批量样本的特征和标签\n",
    "    # 迭代每一批X, y\n",
    "    for X, y in data_iter(batch_size, features, labels):\n",
    "        # 对损失函数l加一个with record，可以记录梯度\n",
    "        with autograd.record():\n",
    "            l = loss(net(X, w, b), y)  # 损失函数是关于w和b的，要对w和b求导\n",
    "        # 开始对损失函数l求梯度，这里l输出是向量，会对l中元素求和得到新变量，再求梯度\n",
    "        # 变量l是长度为batch_size的一维NDArray，\n",
    "        # 执行l.backward()等价于执行l.sum().backward()\n",
    "        l.backward()\n",
    "        # 开始用sgd更新模型参数\n",
    "        sgd([w, b], lr, batch_size)\n",
    "    # 对一轮epoch求完梯度更新完参数后，用训练好的参数拟合预测值，\n",
    "    # 打印出预测值和真实值的loss,输出train_l是（样本数，1），打印出均值\n",
    "    train_l = loss(net(features, w, b), labels)\n",
    "    print('epoch %d, loss %f' % (epoch + 1, train_l.mean().asnumpy()))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-05-09T09:13:50.379280Z",
     "start_time": "2019-05-09T09:13:50.374286Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "([2, -3.4], \n",
       " [[ 1.9994836]\n",
       "  [-3.3999815]]\n",
       " <NDArray 2x1 @cpu(0)>)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "true_w, w"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-05-09T09:13:50.399219Z",
     "start_time": "2019-05-09T09:13:50.382268Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(4.2, \n",
       " [4.19948]\n",
       " <NDArray 1 @cpu(0)>)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "true_b, b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 练习\n",
    "\n",
    "* 为什么`squared_loss`函数中需要使用`reshape`函数？\n",
    "* 尝试使用不同的学习率，观察损失函数值的下降快慢。\n",
    "* 如果样本个数不能被批量大小整除，`data_iter`函数的行为会有什么变化？"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-05-09T09:13:50.416175Z",
     "start_time": "2019-05-09T09:13:50.400217Z"
    }
   },
   "outputs": [],
   "source": [
    "# 1.为什么squared_loss函数中需要使用reshape函数？\n",
    "# 做如下实验, 发现预测值和真实值shape不一样，用reshape把真实值变得和预测值一样，才能计算\n",
    "X = nd.random.normal(scale=1, shape=(5, 2))\n",
    "w = nd.random.normal(scale=0.01, shape=(2, 1))\n",
    "b = nd.zeros(shape=(1,))\n",
    "y_hat = nd.dot(X, w) + b\n",
    "y = true_w[0] * X[:, 0] + true_w[1] * X[:, 1] + b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-05-09T09:13:50.440115Z",
     "start_time": "2019-05-09T09:13:50.418170Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(\n",
       " [[ 0.00215691]\n",
       "  [ 0.00601629]\n",
       "  [ 0.0230494 ]\n",
       "  [-0.00266011]\n",
       "  [-0.00451494]]\n",
       " <NDArray 5x1 @cpu(0)>, \n",
       " [  0.03938711  -3.1831388  -12.660603     0.8042348    3.9560537 ]\n",
       " <NDArray 5 @cpu(0)>)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_hat, y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-05-09T09:13:50.461054Z",
     "start_time": "2019-05-09T09:13:50.446094Z"
    }
   },
   "outputs": [],
   "source": [
    "# 3.如果样本个数不能被批量大小整除，data_iter函数的行为会有什么变化？\n",
    "# 最后一个batch会小于batch_size"
   ]
  }
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