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"# 多层感知机\n",
"\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",
"## 隐藏层\n",
"\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",
"\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",
"\n",
"具体来说，给定一个小批量样本$\\boldsymbol{X} \\in \\mathbb{R}^{n \\times d}$，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隐藏变量）为$\\boldsymbol{H}$，有$\\boldsymbol{H} \\in \\mathbb{R}^{n \\times h}$。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$\\boldsymbol{W}_h \\in \\mathbb{R}^{d \\times h}$和 $\\boldsymbol{b}_h \\in \\mathbb{R}^{1 \\times h}$，输出层的权重和偏差参数分别为$\\boldsymbol{W}_o \\in \\mathbb{R}^{h \\times q}$和$\\boldsymbol{b}_o \\in \\mathbb{R}^{1 \\times q}$。\n",
"\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$\\boldsymbol{O} \\in \\mathbb{R}^{n \\times q}$的计算为\n",
"\n",
"$$\n",
"\\begin{aligned}\n",
"\\boldsymbol{H} &= \\boldsymbol{X} \\boldsymbol{W}_h + \\boldsymbol{b}_h,\\\\\n",
"\\boldsymbol{O} &= \\boldsymbol{H} \\boldsymbol{W}_o + \\boldsymbol{b}_o,\n",
"\\end{aligned} \n",
"$$\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",
"$$\n",
"\\boldsymbol{O} = (\\boldsymbol{X} \\boldsymbol{W}_h + \\boldsymbol{b}_h)\\boldsymbol{W}_o + \\boldsymbol{b}_o = \\boldsymbol{X} \\boldsymbol{W}_h\\boldsymbol{W}_o + \\boldsymbol{b}_h \\boldsymbol{W}_o + \\boldsymbol{b}_o.\n",
"$$\n",
"\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$\\boldsymbol{W}_h\\boldsymbol{W}_o$，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",
"\n",
"## 激活函数\n",
"\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",
"### ReLU函数\n",
"\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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{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"%matplotlib inline\n",
"import d2ltorch as d2lt\n",
"import torch\n",
"from torch import autograd\n",
"\n",
"def xyplot(x_vals, y_vals, name):\n",
" d2lt.set_figsize(figsize=(5, 2.5))\n",
" d2lt.plt.plot(x_vals.data.numpy(), y_vals.data.numpy())\n",
" d2lt.plt.xlabel('x')\n",
" d2lt.plt.ylabel(name + '(x)')"
]
},
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"cell_type": "markdown",
"metadata": {},
"source": [
"我们接下来通过`tensor`提供的`relu`函数来绘制ReLU函数。可以看到，该激活函数是一个两段线性函数。"
]
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"\n",
"\n",
"\n",
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"x = torch.arange(-8.0, 8.0, 0.1)\n",
"x.requires_grad_()\n",
"y = x.relu()\n",
"xyplot(x, y, 'relu')"
]
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"source": [
"显然，当输入为负数时，ReLU函数的导数为0；当输入为正数时，ReLU函数的导数为1。尽管输入为0时ReLU函数不可导，但是我们可以取此处的导数为0。下面绘制ReLU函数的导数。"
]
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"\n",
"\n",
"\n",
"\n"
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"source": [
"y.backward(torch.ones_like(y))\n",
"xyplot(x, x.grad, 'grad of relu')"
]
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"cell_type": "markdown",
"metadata": {},
"source": [
"### sigmoid函数\n",
"\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。"
]
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"execution_count": 4,
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"\n",
"\n",
"\n",
"\n"
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"# 清空x的梯度\n",
"x.grad.data.zero_()\n",
"y = x.sigmoid()\n",
"xyplot(x, y, 'sigmoid')"
]
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"cell_type": "markdown",
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"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",
"\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。"
]
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"execution_count": 5,
"metadata": {},
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"\n",
"\n",
"\n",
"\n"
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"y.backward(torch.ones_like(y))\n",
"xyplot(x, x.grad, 'grad of sigmoid')"
]
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"cell_type": "markdown",
"metadata": {},
"source": [
"### tanh函数\n",
"\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。"
]
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"execution_count": 6,
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"\n",
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"\n",
"\n"
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"source": [
"x.grad.data.zero_()\n",
"y = x.tanh()\n",
"xyplot(x, y, 'tanh')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"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。"
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"source": [
"y.backward(torch.ones_like(y))\n",
"xyplot(x, x.grad, 'grad of tanh')"
]
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"cell_type": "markdown",
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"source": [
"## 多层感知机\n",
"\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",
"$$\n",
"\\begin{aligned}\n",
"\\boldsymbol{H} &= \\phi(\\boldsymbol{X} \\boldsymbol{W}_h + \\boldsymbol{b}_h),\\\\\n",
"\\boldsymbol{O} &= \\boldsymbol{H} \\boldsymbol{W}_o + \\boldsymbol{b}_o,\n",
"\\end{aligned}\n",
"$$\n",
" \n",
"其中$\\phi$表示激活函数。在分类问题中，我们可以对输出$\\boldsymbol{O}$做softmax运算，并使用softmax回归中的交叉熵损失函数。\n",
"在回归问题中，我们将输出层的输出个数设为1，并将输出$\\boldsymbol{O}$直接提供给线性回归中使用的平方损失函数。\n",
"\n",
"\n",
"\n",
"## 小结\n",
"\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。\n",
"\n",
"\n",
"## 练习\n",
"\n",
"* 应用链式法则，推导出sigmoid函数和tanh函数的导数的数学表达式。\n",
"* 查阅资料，了解其他的激活函数。\n",
"\n",
"\n",
"\n",
"\n",
"## 扫码直达[讨论区](https://discuss.gluon.ai/t/topic/6447)\n",
"\n",
""
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