主要參考wildml的博客所寫，所有的代碼都是python實現，並且沒有使用深度學習的框架，所以對理解RNN可以起到很大的幫助。
一、語言模型
如果一個句子有m個詞，那麼這個句子生成的概率就是：
其即假設下一次詞生成的概率和只和句子前面的詞有關，舉一個例子：How are you，生成的概率可以表示爲：
P(How are you) = P(you)*P(you|How，are) 。
二、數據預處理
語料預處理會去掉一些低頻詞從而控制詞典大小，這裏我們截取前8000個高頻詞彙，低頻詞使用一個統一標識替換（這裏是UNKNOWN_TOKEN），在經過預處理之後每一個詞得到一個編號；爲了學出來哪些詞常常作爲句子開始和句子結束，引入SENTENCE_START和SENTENCE_END兩個特殊字符。具體代碼如下：
vocabulary_size = 8000
unknown_token = "UNKNOWN_TOKEN"
sentence_start_token = "SENTENCE_START"
sentence_end_token = "SENTENCE_END"
# Read the data and append SENTENCE_START and SENTENCE_END tokens
print "Reading CSV file..."
with open('data/reddit-comments-2015-08.csv', 'rb') as f:
reader = csv.reader(f, skipinitialspace=True)
reader.next()
# Split full comments into sentences
sentences = itertools.chain(*[nltk.sent_tokenize(x[0].decode('utf-8').lower()) for x in reader])
# Append SENTENCE_START and SENTENCE_END
sentences = ["%s %s %s" % (sentence_start_token, x, sentence_end_token) for x in sentences]
print "Parsed %d sentences." % (len(sentences))
# Tokenize the sentences into words
tokenized_sentences = [nltk.word_tokenize(sent) for sent in sentences]
# Count the word frequencies
word_freq = nltk.FreqDist(itertools.chain(*tokenized_sentences))
print "Found %d unique words tokens." % len(word_freq.items())
# Get the most common words and build index_to_word and word_to_index vectors
vocab = word_freq.most_common(vocabulary_size-1)
index_to_word = [x[0] for x in vocab]
index_to_word.append(unknown_token)
word_to_index = dict([(w,i) for i,w in enumerate(index_to_word)])
print "Using vocabulary size %d." % vocabulary_size
print "The least frequent word in our vocabulary is '%s' and appeared %d times." % (vocab[-1][0], vocab[-1][1])
# Replace all words not in our vocabulary with the unknown token
for i, sent in enumerate(tokenized_sentences):
tokenized_sentences[i] = [w if w in word_to_index else unknown_token for w in sent]
print "\nExample sentence: '%s'" % sentences[0]
print "\nExample sentence after Pre-processing: '%s'" % tokenized_sentences[0]
# Create the training data
X_train = np.asarray([[word_to_index[w] for w in sent[:-1]] for sent in tokenized_sentences])
y_train = np.asarray([[word_to_index[w] for w in sent[1:]] for sent in tokenized_sentences])
Here’s an actual training example from our text:
x:
SENTENCE_START what are n't you understanding about this ? !
[0, 51, 27, 16, 10, 856, 53, 25, 34, 69]
y:
what are n't you understanding about this ? ! SENTENCE_END
[51, 27, 16, 10, 856, 53, 25, 34, 69, 1]循環神經網絡的結構如下圖：
RNN網絡有狀態的概念。如上圖，t表示的是狀態， xt表示的狀態t的輸入， st 表示狀態t時隱層的輸出， ot表示輸出。特別的地方在於，隱層的輸入有兩個來源，一個是當前的 xt輸入、一個是上一個狀態隱層的輸出 st−1。W,U,V 爲參數。使用公式可以將上面結構表示爲：
參數的初始化有很多種方法，都初始化爲0將會導致symmetric calculations ，如何初始化其實是和具體的激活函數有關係，我們這裏使用的是tanh，一種推薦的方式是初始化爲 [−1/√n,1/√n],其中n是前一層接入的鏈接數。
class RNNNumpy:
def __init__(self, word_dim, hidden_dim=100, bptt_truncate=4):
# Assign instance variables
self.word_dim = word_dim
self.hidden_dim = hidden_dim
self.bptt_truncate = bptt_truncate
# Randomly initialize the network parameters
self.U = np.random.uniform(-np.sqrt(1./word_dim), np.sqrt(1./word_dim), (hidden_dim, word_dim))
self.V = np.random.uniform(-np.sqrt(1./hidden_dim), np.sqrt(1./hidden_dim), (word_dim, hidden_dim))
self.W = np.random.uniform(-np.sqrt(1./hidden_dim), np.sqrt(1./hidden_dim), (hidden_dim, hidden_dim))前向傳播代碼如下：
def forward_propagation(self, x):
# The total number of time steps
T = len(x)
# During forward propagation we save all hidden states in s because need them later.
# We add one additional element for the initial hidden, which we set to 0
s = np.zeros((T + 1, self.hidden_dim))
s[-1] = np.zeros(self.hidden_dim)
# The outputs at each time step. Again, we save them for later.
o = np.zeros((T, self.word_dim))
# For each time step...
for t in np.arange(T):
# Note that we are indxing U by x[t]. This is the same as multiplying U with a one-hot vector.
s[t] = np.tanh(self.U[:,x[t]] + self.W.dot(s[t-1]))
o[t] = softmax(self.V.dot(s[t]))
return [o, s]def predict(self, x):
# Perform forward propagation and return index of the highest score
o, s = self.forward_propagation(x)
return np.argmax(o, axis=1)使用交叉熵作爲損失函數，如果有N個樣本，損失函數可以寫爲：
損失函數計算代碼：
def calculate_total_loss(self, x, y):
L = 0
# For each sentence...
for i in np.arange(len(y)):
o, s = self.forward_propagation(x[i])
# We only care about our prediction of the "correct" words
correct_word_predictions = o[np.arange(len(y[i])), y[i]]
# Add to the loss based on how off we were
L += -1 * np.sum(np.log(correct_word_predictions))
return L
def calculate_loss(self, x, y):
# Divide the total loss by the number of training examples
N = np.sum((len(y_i) for y_i in y))
return self.calculate_total_loss(x,y)/NBPTT（Backpropagation Through Time）是一種非常直觀的方法，和傳統的BP類似，只不過傳播的路徑是個循環，並且路徑上的參數是共享的。損失是交叉熵，損失可以表示爲：
其中 yt是真實值， (̂yt)是預估值，將誤差展開可以用圖表示爲：
BPTT梯度更新的代碼爲：
def bptt(self, x, y):
T = len(y)
# Perform forward propagation
o, s = self.forward_propagation(x)
# We accumulate the gradients in these variables
dLdU = np.zeros(self.U.shape)
dLdV = np.zeros(self.V.shape)
dLdW = np.zeros(self.W.shape)
delta_o = o
delta_o[np.arange(len(y)), y] -= 1.
# For each output backwards...
for t in np.arange(T)[::-1]:
dLdV += np.outer(delta_o[t], s[t].T)
# Initial delta calculation: dL/dz
delta_t = self.V.T.dot(delta_o[t]) * (1 - (s[t] ** 2))
# Backpropagation through time (for at most self.bptt_truncate steps)
for bptt_step in np.arange(max(0, t-self.bptt_truncate), t+1)[::-1]:
# print "Backpropagation step t=%d bptt step=%d " % (t, bptt_step)
# Add to gradients at each previous step
dLdW += np.outer(delta_t, s[bptt_step-1])
dLdU[:,x[bptt_step]] += delta_t
# Update delta for next step dL/dz at t-1
delta_t = self.W.T.dot(delta_t) * (1 - s[bptt_step-1] ** 2)
return [dLdU, dLdV, dLdW]tanh和sigmoid函數和導數的取值返回如下圖，可以看到導數取值是[0-1]，用幾次鏈式法則就會將梯度指數級別縮小，所以傳播不了幾層就會出現梯度非常弱。克服這個問題的LSTM是一種最近比較流行的解決方案。
八、Gradient Checking
梯度檢驗是非常有用的，檢查的原理是一個點的梯度等於這個點的斜率，估算一個點的斜率可以通過求極限的方式：
通過比較斜率和梯度的值，我們就可以判斷梯度計算的是否有問題。需要注意的是這個檢驗成本還是很高的，因爲我們的參數個數是百萬量級的。梯度檢驗的代碼：
def gradient_check(self, x, y, h=0.001, error_threshold=0.01):
# Calculate the gradients using backpropagation. We want to checker if these are correct.
bptt_gradients = self.bptt(x, y)
# List of all parameters we want to check.
model_parameters = ['U', 'V', 'W']
# Gradient check for each parameter
for pidx, pname in enumerate(model_parameters):
# Get the actual parameter value from the mode, e.g. model.W
parameter = operator.attrgetter(pname)(self)
print "Performing gradient check for parameter %s with size %d." % (pname, np.prod(parameter.shape))
# Iterate over each element of the parameter matrix, e.g. (0,0), (0,1), ...
it = np.nditer(parameter, flags=['multi_index'], op_flags=['readwrite'])
while not it.finished:
ix = it.multi_index
# Save the original value so we can reset it later
original_value = parameter[ix]
# Estimate the gradient using (f(x+h) - f(x-h))/(2*h)
parameter[ix] = original_value + h
gradplus = self.calculate_total_loss([x],[y])
parameter[ix] = original_value - h
gradminus = self.calculate_total_loss([x],[y])
estimated_gradient = (gradplus - gradminus)/(2*h)
# Reset parameter to original value
parameter[ix] = original_value
# The gradient for this parameter calculated using backpropagation
backprop_gradient = bptt_gradients[pidx][ix]
# calculate The relative error: (|x - y|/(|x| + |y|))
relative_error = np.abs(backprop_gradient - estimated_gradient)/(np.abs(backprop_gradient) + np.abs(estimated_gradient))
# If the error is to large fail the gradient check
if relative_error > error_threshold:
print "Gradient Check ERROR: parameter=%s ix=%s" % (pname, ix)
print "+h Loss: %f" % gradplus
print "-h Loss: %f" % gradminus
print "Estimated_gradient: %f" % estimated_gradient
print "Backpropagation gradient: %f" % backprop_gradient
print "Relative Error: %f" % relative_error
return
it.iternext()
print "Gradient check for parameter %s passed." % (pname)九、SGD實現
W=W−λΔW，其中 ΔW就是梯度，具體代碼：
# Performs one step of SGD.
def numpy_sdg_step(self, x, y, learning_rate):
# Calculate the gradients
dLdU, dLdV, dLdW = self.bptt(x, y)
# Change parameters according to gradients and learning rate
self.U -= learning_rate * dLdU
self.V -= learning_rate * dLdV
self.W -= learning_rate * dLdW十、文本生成
生成過程其實就是模型的應用過程，只需要反覆執行預測函數即可：
def generate_sentence(model):
# We start the sentence with the start token
new_sentence = [word_to_index[sentence_start_token]]
# Repeat until we get an end token
while not new_sentence[-1] == word_to_index[sentence_end_token]:
next_word_probs = model.forward_propagation(new_sentence)
sampled_word = word_to_index[unknown_token]
# We don't want to sample unknown words
while sampled_word == word_to_index[unknown_token]:
samples = np.random.multinomial(1, next_word_probs[-1])
sampled_word = np.argmax(samples)
new_sentence.append(sampled_word)
sentence_str = [index_to_word[x] for x in new_sentence[1:-1]]
return sentence_str
num_sentences = 10
senten_min_length = 7
for i in range(num_sentences):
sent = []
# We want long sentences, not sentences with one or two words
while len(sent) < senten_min_length:
sent = generate_sentence(model)
print " ".join(sent)參考：