引言在许多自然语言处理任务中，许多单词表达是由他们的tf-idf分数决定的。即使这些分数告诉我们一个单词在一个文本中的相对重要性，但是他们并没有告诉我们单词的语义。Word2vec是一类神经网络模型——在给定无标签的语料库的情况下，为语料库中的单词产生一个能表达语义的向量。这些向量通常是有用的：
- 通过词向量来计算两个单词的语义相似性
- 对某些监督型NLP任务如文本分类，语义分析构造特征
接下来我将描述Word2vec其中一个模型，叫做skip-gram模型
skip-gram模型在我详细介绍skip-gram模型前，我们先来了解下训练数据的格式。skip-gram模型的输入是一个单词<span class="MathJax" id="MathJax-Element-1-Frame" tabindex="0" data-mathml="wI" role="presentation" style="box-sizing: border-box; outline: 0px; display: inline; line-height: normal; text-align: left; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; word-break: break-all; position: relative;">wIwI，它的输出是<span class="MathJax" id="MathJax-Element-2-Frame" tabindex="0" data-mathml="wI" role="presentation" style="box-sizing: border-box; outline: 0px; display: inline; line-height: normal; text-align: left; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; word-break: break-all; position: relative;">wIwI的上下文<span class="MathJax" id="MathJax-Element-3-Frame" tabindex="0" data-mathml="wO,1,...,wO,C" role="presentation" style="box-sizing: border-box; outline: 0px; display: inline; line-height: normal; text-align: left; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; word-break: break-all; position: relative;">wO,1,...,wO,CwO,1,...,wO,C，上下文的窗口大小为<span class="MathJax" id="MathJax-Element-4-Frame" tabindex="0" data-mathml="C" role="presentation" style="box-sizing: border-box; outline: 0px; display: inline; line-height: normal; text-align: left; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; word-break: break-all; position: relative;">CC。举个例子，这里有个句子“I drive my car to the store”。我们如果把”car”作为训练输入数据，单词组{“I”, “drive”, “my”, “to”, “the”, “store”}就是输出。所有这些单词，我们会进行one-hot编码。skip-gram模型图如下所示：
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前向传播接下来我们来看下skip-gram神经网络模型，skip-gram的神经网络模型是从前馈神经网络模型改进而来，说白了就是在前馈神经网络模型的基础上，通过一些技巧使得模型更有效。我们先上图，看一波skip-gram的神经网络模型：
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在上图中，输入向量<span class="MathJax" id="MathJax-Element-5-Frame" tabindex="0" data-mathml="x" role="presentation" style="box-sizing: border-box; outline: 0px; display: inline; line-height: normal; text-align: left; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; word-break: break-all; position: relative;">xx代表某个单词的one-hot编码，对应的输出向量{<span class="MathJax" id="MathJax-Element-6-Frame" tabindex="0" data-mathml="y1" role="presentation" style="box-sizing: border-box; outline: 0px; display: inline; line-height: normal; text-align: left; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; word-break: break-all; position: relative;">y1y1,…,<span class="MathJax" id="MathJax-Element-7-Frame" tabindex="0" data-mathml="yC" role="presentation" style="box-sizing: border-box; outline: 0px; display: inline; line-height: normal; text-align: left; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; word-break: break-all; position: relative;">yCyC}。输入层与隐藏层之间的权重矩阵<span class="MathJax" id="MathJax-Element-8-Frame" tabindex="0" data-mathml="W" role="presentation" style="box-sizing: border-box; outline: 0px; display: inline; line-height: normal; text-align: left; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; word-break: break-all; position: relative;">WW的第<span class="MathJax" id="MathJax-Element-9-Frame" tabindex="0" data-mathml="i" role="presentation" style="box-sizing: border-box; outline: 0px; display: inline; line-height: normal; text-align: left; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; word-break: break-all; position: relative;">ii行代表词汇表中第<span class="MathJax" id="MathJax-Element-10-Frame" tabindex="0" data-mathml="i" role="presentation" style="box-sizing: border-box; outline: 0px; display: inline; line-height: normal; text-align: left; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; word-break: break-all; position: relative;">ii个单词的权重。接下来重点来了：这个权重矩阵<span class="MathJax" id="MathJax-Element-11-Frame" tabindex="0" data-mathml="W" role="presentation" style="box-sizing: border-box; outline: 0px; display: inline; line-height: normal; text-align: left; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; word-break: break-all; position: relative;">WW就是我们需要学习的目标（同<span class="MathJax" id="MathJax-Element-12-Frame" tabindex="0" data-mathml="W′" role="presentation" style="box-sizing: border-box; outline: 0px; display: inline; line-height: normal; text-align: left; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; word-break: break-all; position: relative;">W′W′），因为这个权重矩阵包含了词汇表中所有单词的权重信息。上述模型中，每个输出单词向量也有个<span class="MathJax" id="MathJax-Element-13-Frame" tabindex="0" data-mathml="N×V" role="presentation" style="box-sizing: border-box; outline: 0px; display: inline; line-height: normal; text-align: left; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; word-break: break-all; position: relative;">N×VN×V维的输出向量<span class="MathJax" id="MathJax-Element-14-Frame" tabindex="0" data-mathml="W′" role="presentation" style="box-sizing: border-box; outline: 0px; display: inline; line-height: normal; text-align: left; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; word-break: break-all; position: relative;">W′W′。最后模型还有<span class="MathJax" id="MathJax-Element-15-Frame" tabindex="0" data-mathml="N" role="presentation" style="box-sizing: border-box; outline: 0px; display: inline; line-height: normal; text-align: left; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; word-break: break-all; position: relative;">NN个结点的隐藏层，我们可以发现隐藏层节点<span class="MathJax" id="MathJax-Element-16-Frame" tabindex="0" data-mathml="hi" role="presentation" style="box-sizing: border-box; outline: 0px; display: inline; line-height: normal; text-align: left; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; word-break: break-all; position: relative;">hihi的输入就是输入层输入的加权求和。因此由于输入向量<span class="MathJax" id="MathJax-Element-17-Frame" tabindex="0" data-mathml="x" role="presentation" style="box-sizing: border-box; outline: 0px; display: inline; line-height: normal; text-align: left; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; word-break: break-all; position: relative;">xx是one-hot编码，那么只有向量中的非零元素才能对隐藏层产生输入。因此对于输入向量<span class="MathJax" id="MathJax-Element-18-Frame" tabindex="0" data-mathml="x" role="presentation" style="box-sizing: border-box; outline: 0px; display: inline; line-height: normal; text-align: left; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; word-break: break-all; position: relative;">xx其中<span class="MathJax" id="MathJax-Element-19-Frame" tabindex="0" data-mathml="xk=1" role="presentation" style="box-sizing: border-box; outline: 0px; display: inline; line-height: normal; text-align: left; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; word-break: break-all; position: relative;">xk=1xk=1并且<span class="MathJax" id="MathJax-Element-20-Frame" tabindex="0" data-mathml="xk′=0,k≠k′" role="presentation" style="box-sizing: border-box; outline: 0px; display: inline; line-height: normal; text-align: left; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; word-break: break-all; position: relative;">xk′=0,k≠k′xk′=0,k≠k′。所以隐藏层的输出只与权重矩阵第<span class="MathJax" id="MathJax-Element-21-Frame" tabindex="0" data-mathml="k" role="presentation" style="box-sizing: border-box; outline: 0px; display: inline; line-height: normal; text-align: left; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; word-break: break-all; position: relative;">kk行相关，从数学上证明如下：
h=xTW=Wk,.:=vwI(1)(1)h=xTW=Wk,.:=vwI
注意因为输入时one-hot编码，所以这里是不需要使用激活函数的。同理，模型输出结点C×VC×V的输入也是由对应输入结点的加权求和计算得到：
uc,j=v′Twjh(2)(2)uc,j=vwj′Th
其实从上图我们也看到了输出层中的每个单词都是共享权重的，因此我们有uc,j=ujuc,j=uj。最终我们通过softmax函数产生第CC个单词的多项式分布。
p(wc,j=wO,c|wI)=yc,j=exp(uc,j)∑Vj′=1exp(uj′)(3)(3)p(wc,j=wO,c|wI)=yc,j=exp(uc,j)∑j′=1Vexp(uj′)
说白了，这个值就是第C个输出单词的第j个结点的概率大小。
通过BP（反向传播）算法及随机梯度下降来学习权重前面我讲解了skip-gram模型的输入向量及输出的概率表达，以及我们学习的目标。接下来我们详细讲解下学习权重的过程。第一步就是定义损失函数，这个损失函数就是输出单词组的条件概率，一般都是取对数，如下所示：
E=−logp(wO,1,wO,2,...,wO,C|wI)(4)(4)E=−logp(wO,1,wO,2,...,wO,C|wI)
=−log∏c=1Cexp(uc,j)∑Vj′=1exp(u′j)(5)(5)=−log∏c=1Cexp(uc,j)∑j′=1exp(uj′)V
接下来就是对上面的概率求导，具体推导过程可以去看BP算法，我们得到输出权重矩阵W′W′的更新规则：
w′(new)=w′(old)ij−η⋅∑c=1C(yc,j−tc,j)⋅hi(6)(6)w′(new)=wij′(old)−η⋅∑c=1C(yc,j−tc,j)⋅hi
同理权重WW的更新规则如下：
w(new)=w(old)ij−η⋅∑j=1V∑c=1C(yc,j−tc,j)⋅w′ij⋅xj(7)(7)w(new)=wij(old)−η⋅∑j=1V∑c=1C(yc,j−tc,j)⋅wij′⋅xj
从上面的更新规则，我们可以发现，每次更新都需要对整个词汇表求和，因此对于很大的语料库来说，这个计算复杂度是很高的。于是在实际应用中，Google的Mikolov等人提出了分层softmax及负采样可以使得计算复杂度降低很多。
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