3.8. 多层感知机.ipynb

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        "###3.8. 多层感知机\n",
        "我们已经介绍了包括线性回归和softmax回归在内的单层神经网络。然而深度学习主要关注多层模型。在本节中，我们将以多层感知机（multilayer perceptron，MLP）为例，介绍多层神经网络的概念。\n",
        "\n",
        "###3.8.1. 隐藏层\n",
        "多层感知机在单层神经网络的基础上引入了一到多个隐藏层（hidden layer）。隐藏层位于输入层和输出层之间。图3.3展示了一个多层感知机的神经网络图。\n",
        "<div align=center><img src=\"https://zh.gluon.ai/_images/mlp.svg\" width=\"300\"/></div>\n",
        "<center>图 3.3 带有隐藏层的多层感知机。它含有一个隐藏层，该层中有5个隐藏单元</center>\n",
        "\n",
        "\\\n",
        "\n",
        "\n",
        "在图3.3所示的多层感知机中，输入和输出个数分别为4和3，中间的隐藏层中包含了5个隐藏单元（hidden unit）。由于输入层不涉及计算，图3.3中的多层感知机的层数为2。由图3.3可见，隐藏层中的神经元和输入层中各个输入完全连接，输出层中的神经元和隐藏层中的各个神经元也完全连接。因此，多层感知机中的隐藏层和输出层都是全连接层。\\\n",
        "\\\n",
        "具体来说，给定一个小批量样本$X \\in R^{n \\times d}$,其批量大小为$n$,输入个数为d。假设多层感知机只有一个隐藏层，其中隐藏单元个数为$h$.记隐藏层的输出(也称为隐藏层变量或者隐藏变量)为$H$,有$H \\in R^{n \\times h}$。因为隐藏层和输出层均是全连接层，可以设隐藏层的权重参数和偏差参数分别为$W_h \\in R^{d \\times h}$和$b_h \\in R^{1 \\times h}$。输出层的权重和偏差参数分别为$W_o \\in R^{h \\times q}$和$b_o \\in R^{1 \\times q}$。\\\n",
        "我们先来看一种含单隐藏层的多层感知机的设计。其输出 $O \\in R^{n \\times q}$的计算为：\n",
        "$$\n",
        "\\begin{split}\\begin{aligned}\n",
        "\\boldsymbol{H} &= \\boldsymbol{X} \\boldsymbol{W}_h + \\boldsymbol{b}_h,\\\\\n",
        "\\boldsymbol{O} &= \\boldsymbol{H} \\boldsymbol{W}_o + \\boldsymbol{b}_o,\n",
        "\\end{aligned}\\end{split}\n",
        "$$\n",
        "也就是将隐藏层的输出直接作为输出层的输入。如果将以上两个式子联立起来，可以得到\n",
        "$$\n",
        "\\boldsymbol{O} = (\\boldsymbol{X} \\boldsymbol{W}_h + \\boldsymbol{b}_h)\\boldsymbol{W}_o + \\boldsymbol{b}_o = \\boldsymbol{X} \\boldsymbol{W}_h\\boldsymbol{W}_o + \\boldsymbol{b}_h \\boldsymbol{W}_o + \\boldsymbol{b}_o.\n",
        "$$\n",
        "从联立后的式子可以看出，虽然神经网络引入了隐藏层，却依然等价于一个单层神经网络：其中输出层权重参数为\n",
        "$\\boldsymbol{W}_h\\boldsymbol{W}_o$,偏差参数为$\\boldsymbol{b}_h \\boldsymbol{W}_o + \\boldsymbol{b}_o$。不难发现，即便再添加更多的隐藏层，以上设计依然只能与仅含输出层的单层神经网络等价。\n",
        "\\\n",
        "###3.8.2. 激活函数\n",
        "上述问题的根源在于全连接层只是对数据做仿射变换（affine transformation），而多个仿射变换的叠加仍然是一个仿射变换。解决问题的一个方法是引入非线性变换，例如对隐藏变量使用按元素运算的非线性函数进行变换，然后再作为下一个全连接层的输入。这个非线性函数被称为激活函数（activation function）。下面我们介绍几个常用的激活函数。\n",
        "\n",
        "####3.8.2.1. ReLU函数\n",
        "ReLU（rectified linear unit）函数提供了一个很简单的非线性变换。给定元素 x ，该函数定义为\n",
        "$$\n",
        "\\text{ReLU}(x) = \\max(x, 0).\n",
        "$$\n",
        "可以看出，ReLU函数只保留正数元素，并将负数元素清零。为了直观地观察这一非线性变换，我们先定义一个绘图函数xyplot。"
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          "height": 64
        },
        "outputId": "18a310a6-9339-4542-81b6-ba5081e62342"
      },
      "source": [
        "%matplotlib inline\n",
        "from IPython import display\n",
        "from matplotlib import pyplot as plt\n",
        "import tensorflow as tf"
      ],
      "execution_count": 1,
      "outputs": [
        {
          "output_type": "display_data",
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              "<p style=\"color: red;\">\n",
              "The default version of TensorFlow in Colab will soon switch to TensorFlow 2.x.<br>\n",
              "We recommend you <a href=\"https://www.tensorflow.org/guide/migrate\" target=\"_blank\">upgrade</a> now \n",
              "or ensure your notebook will continue to use TensorFlow 1.x via the <code>%tensorflow_version 1.x</code> magic:\n",
              "<a href=\"https://colab.research.google.com/notebooks/tensorflow_version.ipynb\" target=\"_blank\">more info</a>.</p>\n"
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      "source": [
        "def use_svg_display():\n",
        "  #用矢量图显示\n",
        "  display.set_matplotlib_formats('svg')\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"
      ],
      "execution_count": 0,
      "outputs": []
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        "id": "0_UH7DZOFpei",
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      "source": [
        "def xyplot(x_vals,y_vals,name):\n",
        "  set_figsize(figsize=(5,2.5))\n",
        "  plt.plot(x_vals.numpy(),y_vals.numpy())\n",
        "  plt.xlabel('x')\n",
        "  plt.ylabel(name+'(x)')\n",
        "  "
      ],
      "execution_count": 0,
      "outputs": []
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        "outputId": "9ca8bfe9-e6a4-461d-8e31-59a57fda19bd"
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      "source": [
        "x=tf.range(-8.0,8.0,0.1)\n",
        "with tf.GradientTape() as t:\n",
        "  t.watch(x)\n",
        "  y=tf.nn.relu(x)\n",
        "xyplot(x,y,'relu')"
      ],
      "execution_count": 5,
      "outputs": [
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          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "_bq_tMZfGjqQ",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 261
        },
        "outputId": "73575439-c086-47f7-9040-25077e24803b"
      },
      "source": [
        "x_grad=t.gradient(y,x) #求导\n",
        "xyplot(x,x_grad,'grad of relu')"
      ],
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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          "metadata": {
            "tags": []
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        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "KTEe3nqOG4gR",
        "colab_type": "text"
      },
      "source": [
        "####3.8.2.2. sigmoid函数\n",
        "sigmoid函数可以将元素的值变换到0和1之间：\n",
        "$$\n",
        "\\text{sigmoid}(x) = \\frac{1}{1 + \\exp(-x)}.\n",
        "$$\n",
        "sigmoid函数在早期的神经网络中较为普遍，但它目前逐渐被更简单的ReLU函数取代。在后面“循环神经网络”一章中我们会介绍如何利用它值域在0到1之间这一特性来控制信息在神经网络中的流动。下面绘制了sigmoid函数。当输入接近0时，sigmoid函数接近线性变换。\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ONvHLvr6HSoT",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 261
        },
        "outputId": "61fa71cb-795b-4be7-a636-0223c071be02"
      },
      "source": [
        "with tf.GradientTape()as t:\n",
        "  t.watch(x)\n",
        "  y=tf.sigmoid(x)\n",
        "xyplot(x,y,'sigmoid')"
      ],
      "execution_count": 7,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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          "metadata": {
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      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "X1zowx9aHgPW",
        "colab_type": "text"
      },
      "source": [
        "依据链式法则，sigmoid函数的导数\n",
        "$$\n",
        "\\text{sigmoid}'(x) = \\text{sigmoid}(x)\\left(1-\\text{sigmoid}(x)\\right).\n",
        "$$\n",
        "下面绘制了sigmoid函数的导数。当输入为0时，sigmoid函数的导数达到最大值0.25；当输入越偏离0时，sigmoid函数的导数越接近0。"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "3naXKYX0Hqng",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 261
        },
        "outputId": "83d6aae1-d00e-4acc-e5b5-61e6e268abd7"
      },
      "source": [
        "x_grad=t.gradient(y,x)\n",
        "xyplot(x,x_grad,'grad of sigmoid')"
      ],
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
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            "tags": []
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        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "hm-m1nhAHzti",
        "colab_type": "text"
      },
      "source": [
        "####3.8.2.3. tanh函数\n",
        "tanh（双曲正切）函数可以将元素的值变换到-1和1之间：\n",
        "$$\n",
        "\\text{tanh}(x) = \\frac{1 - \\exp(-2x)}{1 + \\exp(-2x)}.\n",
        "$$\n",
        "我们接着绘制tanh函数。当输入接近0时，tanh函数接近线性变换。虽然该函数的形状和sigmoid函数的形状很像，但tanh函数在坐标系的原点上对称。"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "wmJWdlEkH_Wx",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 261
        },
        "outputId": "ca7bb93b-b2c5-45b5-c11a-3efcf0402af9"
      },
      "source": [
        "with tf.GradientTape() as t:\n",
        "  t.watch(x)\n",
        "  y=tf.tanh(x)\n",
        "\n",
        "xyplot(x,y,'tanh')"
      ],
      "execution_count": 9,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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          "metadata": {
            "tags": []
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    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "_bNofNGCIIwZ",
        "colab_type": "text"
      },
      "source": [
        "依据链式法则，tanh函数的导数\n",
        "$$\n",
        "\\text{tanh}'(x) = 1 - \\text{tanh}^2(x).\n",
        "$$\n",
        "下面绘制了tanh函数的导数。当输入为0时，tanh函数的导数达到最大值1；当输入越偏离0时，tanh函数的导数越接近0。"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "PpEmTYpaIS_w",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 261
        },
        "outputId": "4ace0dd5-766f-4208-d1ab-b8dff2e1cae0"
      },
      "source": [
        "x_grad=t.gradient(y,x)\n",
        "xyplot(x,x_grad,'grad of tanh')"
      ],
      "execution_count": 10,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 360x180 with 1 Axes>"
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        "id": "niXvsA52IdaT",
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      "source": [
        "###3.8.3. 多层感知机\n",
        "多层感知机就是含有至少一个隐藏层的由全连接层组成的神经网络，且每个隐藏层的输出通过激活函数进行变换。多层感知机的层数和各隐藏层中隐藏单元个数都是超参数。以单隐藏层为例并沿用本节之前定义的符号，多层感知机按以下方式计算输出：\n",
        "$$\n",
        "\\begin{split}\\begin{aligned}\n",
        "\\boldsymbol{H} &= \\phi(\\boldsymbol{X} \\boldsymbol{W}_h + \\boldsymbol{b}_h),\\\\\n",
        "\\boldsymbol{O} &= \\boldsymbol{H} \\boldsymbol{W}_o + \\boldsymbol{b}_o,\n",
        "\\end{aligned}\\end{split}\n",
        "$$\n",
        "其中$\\phi$表示激活函数。在分类问题中，我们可以对输出 $O$ 做softmax运算，并使用softmax回归中的交叉熵损失函数。 在回归问题中，我们将输出层的输出个数设为1，并将输出 $O$ 直接提供给线性回归中使用的平方损失函数。\n",
        "\n",
        "###3.8.4. 小结\n",
        "多层感知机在输出层与输入层之间加入了一个或多个全连接隐藏层，并通过激活函数对隐藏层输出进行变换。\n",
        "常用的激活函数包括ReLU函数、sigmoid函数和tanh函数。"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "sGg2EKQkI9X_",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        ""
      ],
      "execution_count": 0,
      "outputs": []
    }
  ]
}