条件随机场的代码理解(pytorch)

CRF

本文用来记录学习过程,自己也是一个初学者,如果有什么问题还请指出.

概率图模型

概率图模型可以分为有向图和无向图,其中无向图是联合概率的生成图模型,常见的比如马尔科夫网络。其中给定一组输入随机变量X的条件下,假设输出随机变量Y是马尔科夫随机场,这就是条件随机场(CRF)。这里所说的是线性条件随机场用于序列标注问题,给定输入序列预测输出序列的判别式模型。马尔科夫网络可以被分为若干个最大团的乘积。 P ( Y ) = 1 Z ( x ) ∏ c = 1 n ψ c ( c ) P(Y)=\frac{1}{Z(x)}\prod_{c=1}^n\psi_c(c) P(Y)=Z(x)1c=1nψc(c)
CRF公式可以表示为: P ( y ∣ x ) = 1 z ( x ) e x p ( ∑ i ∑ k λ k t k ( y i − 1 , y i , x , i ) + ∑ i ∑ l μ l s l ( y i , x , i ) ) P(y|x)=\frac{1}{z(x)}exp(\sum_{i}\sum_k\lambda_kt_k(y_{i-1},y_i,x,i)+\sum_{i}\sum_l\mu_ls_l(y_i,x,i)) P(yx)=z(x)1exp(ikλktk(yi1,yi,x,i)+ilμlsl(yi,x,i))

最大化P(Y)得到所需结果

根据所学最大化似然估计可以求出参数,即所求目标转变为最小化 l o g Z ( x ) − ∑ i , k λ k t k ( y i − 1 , y i , x , i ) + ∑ i , l μ l s l ( y i , x , i ) ) logZ(x)-\sum_{i,k}\lambda_kt_k(y_{i-1},y_i,x,i)+\sum_{i,l}\mu_ls_l(y_i,x,i)) logZ(x)i,kλktk(yi1,yi,x,i)+i,lμlsl(yi,x,i))

代码推理

所用的代码来自官方pytorch源码,自己做一些梳理和解释。

  • 训练数据集合为,句子和对应的词性标签
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()
)]
  • 标记句子中的每一个单词,同时也将标签编号
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)#为每一个单词进行编码

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

  • 将句子中的单词按照上面的字典转化为序列
def prepare_sequence(seq, to_ix):
    idxs = [to_ix[w] for w in seq]#将句子转化为序列号
    return torch.tensor(idxs, dtype=torch.long)

  • 将标签也转化为序列
targets = torch.tensor([tag_to_ix[t] for t in tags], dtype=torch.long)
  • 按照上述公式计算最大似然估计,这里的forward_score gold_score分别对应上述公式的俩个部分
loss = model.neg_log_likelihood(sentence_in, targets)
def neg_log_likelihood(self, sentence, tags):
    feats = self._get_lstm_features(sentence)
    forward_score = self._forward_alg(feats)
    gold_score = self._score_sentence(feats, tags)
    return forward_score - gold_score

通过lstm获取句子的特征feats,这里使用的是单层双向LSTM模型提取句子的特征,同时要将其转化为标签的维度,通过一个线性层实现

self.lstm = nn.LSTM(embedding_dim, hidden_dim // 2,
                            num_layers=1, bidirectional=True)
self.hidden2tag = nn.Linear(hidden_dim, self.tagset_size)

先说分子部分也就是后面一部分_score_sentence函数,首先将标签转化为序列,同时将开头标签加入其中,然后根据给定的公式分别求和状态值和转移值,最后将转移到结尾的值加上,注意start和end没有状态值

    def _score_sentence(self, feats, tags):
        # Gives the score of a provided tag sequence
        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

第二部分: Z ( x ) = ∑ s l o g ( e x p ( ∑ i λ k t k ( y i − 1 , y i , x , i ) + ∑ i μ l s l ( y i , x , i ) ) ) Z(x)=\sum_slog(exp(\sum_{i}\lambda_kt_k(y_{i-1},y_i,x,i)+\sum_{i}\mu_ls_l(y_i,x,i))) Z(x)=slog(exp(iλktk(yi1,yi,x,i)+iμlsl(yi,x,i)))
将所有的序列求和,然后求log_sum_exp,因为要避免上溢出,所以等价于求

def log_sum_exp(vec):
    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)))

遍历所有的序列,遍历每一个标签,然后累加转移状态,特征状态,进行logsumexp求和。初始化将start标签特征矩阵置为0其余为-10000

    def _forward_alg(self, feats):
        # Do the forward algorithm to compute the partition function
        init_alphas = torch.full((1, self.tagset_size), -10000.)
        # 第一个标签必然是start标签,得分必然是1
        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

        # Iterate through the sentence
        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
                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)
        terminal_var = forward_var + self.transitions[self.tag_to_ix[STOP_TAG]]
        alpha = log_sum_exp(terminal_var)
        return alpha
  • 接下来对其进行迭代,计算损失,得到最佳参数
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()
  • 计算viterbi解码
遍历每一个序列
	遍历每一个标签
	  计算累加和(这里不包括特征状态值),选择最大得分对应的序号(标签中的序号)保存在bptrs_t,同时将得分保存在viterbivars_t中。
	将得分都加上特征得分,保存在forward_var中
	将bptrs_t列表添加到backpointers用来回溯

添加stop标签转移值,计算出最终得分的最大一个序号,然后根据保存的path_score计算出序号,输出序号
  • 所有的代码
# Author: Robert Guthrie

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

torch.manual_seed(1)

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


def prepare_sequence(seq, to_ix):
    idxs = [to_ix[w] for w in seq]#将句子转化为序列号
    return torch.tensor(idxs, dtype=torch.long)


# Compute log sum exp in a numerically stable way for the forward algorithm
def log_sum_exp(vec):
    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)))



class BiLSTM_CRF(nn.Module):

    def __init__(self, vocab_size, tag_to_ix, embedding_dim, hidden_dim):#单词数量 标签字典 
        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)#标签数量

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

        # Maps the output of the LSTM into tag space.
        self.hidden2tag = nn.Linear(hidden_dim, self.tagset_size)

        # Matrix of transition parameters.  Entry i,j is the score of
        # transitioning *to* i *from* j.
        self.transitions = nn.Parameter(
            torch.randn(self.tagset_size, self.tagset_size))#转移矩阵

        # These two statements enforce the constraint that we never transfer
        # to the start tag and we never transfer from the stop tag
        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):
        return (torch.randn(2, 1, self.hidden_dim // 2),
                torch.randn(2, 1, self.hidden_dim // 2))

    def _forward_alg(self, feats):
        # Do the forward algorithm to compute the partition function
        init_alphas = torch.full((1, self.tagset_size), -10000.)
        # 第一个标签必然是start标签,得分必然是1
        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

        # Iterate through the sentence
        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
                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)
        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):
        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):
        # Gives the score of a provided tag sequence
        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):
        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:#遍历一个句子中的单词数量
            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 = forward_var + self.transitions[next_tag]
                best_tag_id = argmax(next_tag_var)#转移到该标签下的最大值
                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)#取最大的一个值,是第几个位置,确定了它是哪一个标签
        path_score = terminal_var[0][best_tag_id]#最大的值

        # 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):
        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):  # dont confuse this with _forward_alg above.
        # Get the emission scores from the BiLSTM
        lstm_feats = self._get_lstm_features(sentence)#将文字进行lstm编码然后转化成tag_size输出

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

START_TAG = ""
STOP_TAG = ""
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)#为每一个单词进行编码

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!

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