线性链条件随机场(CRF)的原理与实现

基本原理

损失函数

（线性链）CRF通常用于序列标注任务，对于输入序列\(x\)和标签序列\(y\)，定义匹配分数：

\[s(x,y) = \sum_{i=0}^l T(y_i, y_{i+1}) + \sum_{i=1}^l U(x_i, y_i)
\]

这里\(l\)是序列长度，\(T\)和\(U\)都是可学习的参数，\(T(y_i, y_{i+1})\)表示第\(i\)步的标签是\(y_i\)，第\(i+1\)步标签是\(y_{i+1}\)的转移分数，\(U(x_i,y_i)\)表示第\(i\)步输入\(x_i\)对应的标签是\(y_i\)的发射分数。注意这里在计算转移分数\(T\)时，状态转移链为\(y_0\rightarrow y_1 \rightarrow \dots \rightarrow y_l \rightarrow y_{l+1}\)，因为人为地加入了START_TAG和STOP_TAG标签。

为了解决标注偏置问题，CRF需要做全局归一化，具体而言就是输入\(x\)对应的标签序列为\(y\)的概率定义为：

\[P(y|x)=\frac{e^{s(x,y)}}{Z(x)} = \frac{e^{s(x,y)}}{\sum_{\tilde{y}\in Y_x}e^{s(x,y)}}
\]

因此这里最麻烦的就是计算配分函数(partition function)\(Z(x)\)，因为它要遍历所有路径。

在训练过程中，我们希望最大化正确标签序列的对数概率，即：

\[\log P(y|x)=\log P(\frac{e^{s(x,y)}}{Z(x)}) = s(x,y) - \log Z(x) = s(x,y) - \log (\sum_{\tilde{y}\in Y_x}e^{s(x,y)})
\]

也就是最小化负对数似然，即损失函数为：

\[-\log P(y|x)=\log P(\frac{e^{s(x,y)}}{Z(x)}) = \log (\sum_{\tilde{y}\in Y_x}e^{s(x,y)}) - s(x,y)
\]

配分函数计算

接下来我们来讨论怎么计算\(Z(x)\)。我们使用前向算法计算\(Z(x)\)，伪码如下：

1. 初始化，对于\(y_2\)的所有取值\(y_2^*\)，定义

\[\alpha_1(y_2^*) = \sum_{y_1^*} \exp(U(x_1, y_1^*) + T(y_1^*, y_2^*))
\]

这里\(y_k\)表示\(k\)时刻的标签，它的取值空间是标签控件，如B,I,O等，某一个具体的取值记为\(y_k^*\)。\(\alpha_k(y_{k+1}^*)\)可以认为是时刻\(k\)时的非规范化概率。注意这里\(y_{k+1}^*\)我们只用了一个标签，其实我们要在整个标签空间遍历，对于\(y_{k+1}\)的每一个取值都算一遍。

2. 对于\(k = 2, 3, \dots, l-1\)以及\(y_{k+1}\)的所有取值\(y_{k+1}^*\)，都有：

\[\log (\alpha_k(y_{k+1}^*)) = \log \sum_{y_k^*}\exp \left(U(x_k, y_k^*)+T(y_k^*, y_{k+1}^*) + \log(\alpha_{k-1}(y_k^*)) \right)
\]

这里\(y_k\)和\(y_{k+1}\)都是一个具体的取值，这意味着这一步的计算复杂度是\(O(N^2)\)的，其中\(N\)是标签数目。

3. 最终：

\[Z(x) = \sum_{y_l^*} \exp \left(U(x_l, y_l^*) + \log(\alpha_{l-1}(y_l^*)) \right)
\]

注意到伪码第二步就是所谓的logsumexp，这可能会导致问题。因为如果求指数特别大，可能会导致溢出。因此这里存在一个小trick使得计算时数值稳定：

\[\log \sum_k \exp(z_k) = \max (\mathbf{z}) + \log \sum_k \exp(z_k - \max(\mathbf{z}))
\]

证明如下：

\[\log \sum_k \exp(z_k) = \log \sum_k (\exp(z_k -c) \cdot \exp(c)) = \log[\exp(c) \cdot \sum_k \exp(z_k -c)] = c + \log \sum_k \exp(z_k -c) \qquad \text{令} \ c = \max({\mathbf{z}})
\]

代码实现

以下代码参考Pytorch关于Bi-LSTM+CRF的tutorial。首先导入需要的模块：

import torch
import torch.autograd as autograd
import torch.nn as nn
import torch.optim as optim

torch.manual_seed(1)

为了使模型易读，定义几个辅助函数：

def argmax(vec):
    """return the argmax as a python int"""
    _, idx = torch.max(vec, 1)
    return idx.item()

def prepare_sequence(seq, to_ix):
    """word2id"""
    idxs = [to_ix[w] for w in seq]
    return torch.tensor(idxs, dtype=torch.long)

def log_sum_exp(vec):
    """Compute log sum exp in a numerically stable way for the forward algorithm
    这个函数在Pytorch和TensorFlow其实都有，这里作者为了讲解又实现了一次
    """
    max_score = vec[0, argmax(vec)]
    max_score_broadcast = max_score.view(1, -1).expand(1, vec.size()[1])
    return max_score + \
        torch.log(torch.sum(torch.exp(vec - max_score_broadcast)))

接下来定义整个模型：

class BiLSTM_CRF(nn.Module):

    def __init__(self, vocab_size, tag_to_ix, embedding_dim, hidden_dim):
        super(BiLSTM_CRF, self).__init__()
        self.embedding_dim = embedding_dim
        self.hidden_dim = hidden_dim
        self.vocab_size = vocab_size
        self.tag_to_ix = tag_to_ix
        self.tagset_size = len(tag_to_ix)

        self.word_embeds = nn.Embedding(vocab_size, embedding_dim)
        self.lstm = nn.LSTM(embedding_dim, hidden_dim // 2,
                            num_layers=1, bidirectional=True)

        # 将LSTM的输出映射到标签空间
        # 相当于公式中的发射矩阵U
        self.hidden2tag = nn.Linear(hidden_dim, self.tagset_size)

        # 转移矩阵，从标签i转移到标签j的分数
        # tagset_size包含了人为加入的START_TAG和STOP_TAG
        self.transitions = nn.Parameter(
            torch.randn(self.tagset_size, self.tagset_size))

        # 下面这两个约束不能转移到START_TAG，也不能从STOP_TAG开始转移
        self.transitions.data[tag_to_ix[START_TAG], :] = -10000
        self.transitions.data[:, tag_to_ix[STOP_TAG]] = -10000

        self.hidden = self.init_hidden()

    def init_hidden(self):
        """初始化LSTM"""
        return (torch.randn(2, 1, self.hidden_dim // 2),
                torch.randn(2, 1, self.hidden_dim // 2))

    def _forward_alg(self, feats):
        """计算配分函数Z(x)"""

        # 对应于伪码第一步
        init_alphas = torch.full((1, self.tagset_size), -10000.)
        # START_TAG has all of the score.
        init_alphas[0][self.tag_to_ix[START_TAG]] = 0.

        # Wrap in a variable so that we will get automatic backprop
        forward_var = init_alphas

        # 对应于伪码第二步的循环，迭代整个句子
        for feat in feats:
            alphas_t = []  # The forward tensors at this timestep
            for next_tag in range(self.tagset_size):
                # broadcast the emission score: it is the same regardless of
                # the previous tag
                emit_score = feat[next_tag].view(
                    1, -1).expand(1, self.tagset_size)
                # the ith entry of trans_score is the score of transitioning to
                # next_tag from i
                trans_score = self.transitions[next_tag].view(1, -1)
                # The ith entry of next_tag_var is the value for the
                # edge (i -> next_tag) before we do log-sum-exp
                # 这里对应了伪码第二步中三者求和
                next_tag_var = forward_var + trans_score + emit_score
                # The forward variable for this tag is log-sum-exp of all the scores.
                alphas_t.append(log_sum_exp(next_tag_var).view(1))
            forward_var = torch.cat(alphas_t).view(1, -1)
        # 对应于伪码第三步，注意损失函数最终是要logZ(x)，所以又是一个logsumexp
        terminal_var = forward_var + self.transitions[self.tag_to_ix[STOP_TAG]]
        alpha = log_sum_exp(terminal_var)
        return alpha

    def _get_lstm_features(self, sentence):
        """调用LSTM获得每个token的隐状态，这里可以替换为任意的特征函数，
        LSTM返回的特征就是公式中的x
        """
        self.hidden = self.init_hidden()
        embeds = self.word_embeds(sentence).view(len(sentence), 1, -1)
        lstm_out, self.hidden = self.lstm(embeds, self.hidden)
        lstm_out = lstm_out.view(len(sentence), self.hidden_dim)
        lstm_feats = self.hidden2tag(lstm_out)
        return lstm_feats

    def _score_sentence(self, feats, tags):
        """计算给定输入序列和标签序列的匹配函数，即公式中的s函数"""
        score = torch.zeros(1)
        tags = torch.cat([torch.tensor([self.tag_to_ix[START_TAG]], dtype=torch.long), tags])
        for i, feat in enumerate(feats):
            score = score + \
                self.transitions[tags[i + 1], tags[i]] + feat[tags[i + 1]]
        score = score + self.transitions[self.tag_to_ix[STOP_TAG], tags[-1]]
        return score

    def _viterbi_decode(self, feats):
        """维特比解码，给定输入x和相关参数(发射矩阵和转移矩阵)，或者概率最大的标签序列
        """
        backpointers = []

        # Initialize the viterbi variables in log space
        init_vvars = torch.full((1, self.tagset_size), -10000.)
        init_vvars[0][self.tag_to_ix[START_TAG]] = 0

        # forward_var at step i holds the viterbi variables for step i-1
        forward_var = init_vvars
        for feat in feats:
            bptrs_t = []  # holds the backpointers for this step
            viterbivars_t = []  # holds the viterbi variables for this step

            for next_tag in range(self.tagset_size):
                # next_tag_var[i] holds the viterbi variable for tag i at the
                # previous step, plus the score of transitioning
                # from tag i to next_tag.
                # We don't include the emission scores here because the max
                # does not depend on them (we add them in below)
                next_tag_var = forward_var + self.transitions[next_tag]
                best_tag_id = argmax(next_tag_var)
                bptrs_t.append(best_tag_id)
                viterbivars_t.append(next_tag_var[0][best_tag_id].view(1))
            # Now add in the emission scores, and assign forward_var to the set
            # of viterbi variables we just computed
            forward_var = (torch.cat(viterbivars_t) + feat).view(1, -1)
            backpointers.append(bptrs_t)

        # Transition to STOP_TAG
        terminal_var = forward_var + self.transitions[self.tag_to_ix[STOP_TAG]]
        best_tag_id = argmax(terminal_var)
        path_score = terminal_var[0][best_tag_id]

        # Follow the back pointers to decode the best path.
        best_path = [best_tag_id]
        for bptrs_t in reversed(backpointers):
            best_tag_id = bptrs_t[best_tag_id]
            best_path.append(best_tag_id)
        # Pop off the start tag (we dont want to return that to the caller)
        start = best_path.pop()
        assert start == self.tag_to_ix[START_TAG]  # Sanity check
        best_path.reverse()
        return path_score, best_path

    def neg_log_likelihood(self, sentence, tags):
        """损失函数 = Z(x) - s(x,y)
        """
        feats = self._get_lstm_features(sentence)
        forward_score = self._forward_alg(feats)
        gold_score = self._score_sentence(feats, tags)
        return forward_score - gold_score

    def forward(self, sentence):
        """预测函数，注意这个函数和_forward_alg不一样
        这里给定一个句子，预测最有可能的标签序列
        """
        # Get the emission scores from the BiLSTM
        lstm_feats = self._get_lstm_features(sentence)

        # Find the best path, given the features.
        score, tag_seq = self._viterbi_decode(lstm_feats)
        return score, tag_seq

最后，把上述模型拼起来得到一个完整的可运行实例，这里就不再讲解：

START_TAG = "<START>"
STOP_TAG = "<STOP>"
EMBEDDING_DIM = 5
HIDDEN_DIM = 4

# Make up some training data
training_data = [(
    "the wall street journal reported today that apple corporation made money".split(),
    "B I I I O O O B I O O".split()
), (
    "georgia tech is a university in georgia".split(),
    "B I O O O O B".split()
)]

word_to_ix = {}
for sentence, tags in training_data:
    for word in sentence:
        if word not in word_to_ix:
            word_to_ix[word] = len(word_to_ix)

tag_to_ix = {"B": 0, "I": 1, "O": 2, START_TAG: 3, STOP_TAG: 4}

model = BiLSTM_CRF(len(word_to_ix), tag_to_ix, EMBEDDING_DIM, HIDDEN_DIM)
optimizer = optim.SGD(model.parameters(), lr=0.01, weight_decay=1e-4)

# Check predictions before training
with torch.no_grad():
    precheck_sent = prepare_sequence(training_data[0][0], word_to_ix)
    precheck_tags = torch.tensor([tag_to_ix[t] for t in training_data[0][1]], dtype=torch.long)
    print(model(precheck_sent))

# Make sure prepare_sequence from earlier in the LSTM section is loaded
for epoch in range(
        300):  # again, normally you would NOT do 300 epochs, it is toy data
    for sentence, tags in training_data:
        # Step 1. Remember that Pytorch accumulates gradients.
        # We need to clear them out before each instance
        model.zero_grad()

        # Step 2. Get our inputs ready for the network, that is,
        # turn them into Tensors of word indices.
        sentence_in = prepare_sequence(sentence, word_to_ix)
        targets = torch.tensor([tag_to_ix[t] for t in tags], dtype=torch.long)

        # Step 3. Run our forward pass.
        loss = model.neg_log_likelihood(sentence_in, targets)

        # Step 4. Compute the loss, gradients, and update the parameters by
        # calling optimizer.step()
        loss.backward()
        optimizer.step()

# Check predictions after training
with torch.no_grad():
    precheck_sent = prepare_sequence(training_data[0][0], word_to_ix)
    print(model(precheck_sent))
# We got it!

参考资料

[1]. https://towardsdatascience.com/implementing-a-linear-chain-conditional-random-field-crf-in-pytorch-16b0b9c4b4ea

[2]. https://zhuanlan.zhihu.com/p/27338210