基本原理
損失函數
（線性鏈）CRF通常用於序列標注任務，對於輸入序列\(x\)和標簽序列\(y\)，定義匹配分數：
\[s(x,y) = \sum_{i=0}^l T(y_i, y_{i+1}) + \sum_{i=1}^l U(x_i, y_i) \]
這里\(l\)是序列長度，\(T\)和\(U\)都是可學習的參數，\(T(y_i, y_{i+1})\)表示第\(i\)步的標簽是\(y_i\)，第\(i+1\)步標簽是\(y_{i+1}\)的轉移分數，\(U(x_i,y_i)\)表示第\(i\)步輸入\(x_i\)對應的標簽是\(y_i\)的發射分數。注意這里在計算轉移分數\(T\)時，狀態轉移鏈為\(y_0\rightarrow y_1 \rightarrow \dots \rightarrow y_l \rightarrow y_{l+1}\)，因為人為地加入了START_TAG和STOP_TAG標簽。
為了解決標注偏置問題，CRF需要做全局歸一化，具體而言就是輸入\(x\)對應的標簽序列為\(y\)的概率定義為：
\[P(y|x)=\frac{e^{s(x,y)}}{Z(x)} = \frac{e^{s(x,y)}}{\sum_{\tilde{y}\in Y_x}e^{s(x,y)}} \]
因此這里最麻煩的就是計算配分函數(partition function)\(Z(x)\)，因為它要遍歷所有路徑。
在訓練過程中，我們希望最大化正確標簽序列的對數概率，即：
\[\log P(y|x)=\log P(\frac{e^{s(x,y)}}{Z(x)}) = s(x,y) - \log Z(x) = s(x,y) - \log (\sum_{\tilde{y}\in Y_x}e^{s(x,y)}) \]
也就是最小化負對數似然，即損失函數為：
\[-\log P(y|x)=\log P(\frac{e^{s(x,y)}}{Z(x)}) = \log (\sum_{\tilde{y}\in Y_x}e^{s(x,y)}) - s(x,y) \]
配分函數計算
接下來我們來討論怎么計算\(Z(x)\)。我們使用前向算法計算\(Z(x)\)，偽碼如下：
- 初始化，對於\(y_2\)的所有取值\(y_2^*\)，定義
\[\alpha_1(y_2^*) = \sum_{y_1^*} \exp(U(x_1, y_1^*) + T(y_1^*, y_2^*)) \]
這里\(y_k\)表示\(k\)時刻的標簽，它的取值空間是標簽控件，如B,I,O等，某一個具體的取值記為\(y_k^*\)。\(\alpha_k(y_{k+1}^*)\)可以認為是時刻\(k\)時的非規范化概率。注意這里\(y_{k+1}^*\)我們只用了一個標簽，其實我們要在整個標簽空間遍歷，對於\(y_{k+1}\)的每一個取值都算一遍。
2. 對於\(k = 2, 3, \dots, l-1\)以及\(y_{k+1}\)的所有取值\(y_{k+1}^*\)，都有：
\[\log (\alpha_k(y_{k+1}^*)) = \log \sum_{y_k^*}\exp \left(U(x_k, y_k^*)+T(y_k^*, y_{k+1}^*) + \log(\alpha_{k-1}(y_k^*)) \right) \]
這里\(y_k\)和\(y_{k+1}\)都是一個具體的取值，這意味着這一步的計算復雜度是\(O(N^2)\)的，其中\(N\)是標簽數目。
3. 最終：
\[Z(x) = \sum_{y_l^*} \exp \left(U(x_l, y_l^*) + \log(\alpha_{l-1}(y_l^*)) \right) \]
注意到偽碼第二步就是所謂的logsumexp，這可能會導致問題。因為如果求指數特別大，可能會導致溢出。因此這里存在一個小trick使得計算時數值穩定：
\[\log \sum_k \exp(z_k) = \max (\mathbf{z}) + \log \sum_k \exp(z_k - \max(\mathbf{z})) \]
證明如下：
\[\log \sum_k \exp(z_k) = \log \sum_k (\exp(z_k -c) \cdot \exp(c)) = \log[\exp(c) \cdot \sum_k \exp(z_k -c)] = c + \log \sum_k \exp(z_k -c) \qquad \text{令} \ c = \max({\mathbf{z}}) \]
代碼實現
以下代碼參考Pytorch關於Bi-LSTM+CRF的tutorial。首先導入需要的模塊：
import torch
import torch.autograd as autograd
import torch.nn as nn
import torch.optim as optim
torch.manual_seed(1)
為了使模型易讀，定義幾個輔助函數：
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
這個函數在Pytorch和TensorFlow其實都有，這里作者為了講解又實現了一次
"""
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)))
接下來定義整個模型：
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)
# 將LSTM的輸出映射到標簽空間
# 相當於公式中的發射矩陣U
self.hidden2tag = nn.Linear(hidden_dim, self.tagset_size)
# 轉移矩陣，從標簽i轉移到標簽j的分數
# tagset_size包含了人為加入的START_TAG和STOP_TAG
self.transitions = nn.Parameter(
torch.randn(self.tagset_size, self.tagset_size))
# 下面這兩個約束不能轉移到START_TAG，也不能從STOP_TAG開始轉移
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):
"""計算配分函數Z(x)"""
# 對應於偽碼第一步
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
# 對應於偽碼第二步的循環，迭代整個句子
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
# 這里對應了偽碼第二步中三者求和
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)
# 對應於偽碼第三步，注意損失函數最終是要logZ(x)，所以又是一個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):
"""調用LSTM獲得每個token的隱狀態，這里可以替換為任意的特征函數，
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):
"""計算給定輸入序列和標簽序列的匹配函數，即公式中的s函數"""
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):
"""維特比解碼，給定輸入x和相關參數(發射矩陣和轉移矩陣)，或者概率最大的標簽序列
"""
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):
"""損失函數 = 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):
"""預測函數，注意這個函數和_forward_alg不一樣
這里給定一個句子，預測最有可能的標簽序列
"""
# 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，把上述模型拼起來得到一個完整的可運行實例，這里就不再講解：
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!
參考資料
[1]. https://towardsdatascience.com/implementing-a-linear-chain-conditional-random-field-crf-in-pytorch-16b0b9c4b4ea
[2]. https://zhuanlan.zhihu.com/p/27338210