【Deep Learning】循环神经网络(RNN)推导和实现

       主要参考wildml的博客所写，所有的代码都是python实现，并且没有使用深度学习的框架，所以对理解RNN可以起到很大的帮助。

一、语言模型

        如果一个句子有m个词，那么这个句子生成的概率就是：

        其即假设下一次词生成的概率和只和句子前面的词有关，举一个例子：How are you，生成的概率可以表示为： 

 P(How are you) = P(you)*P(you|How，are) 。

二、数据预处理

       语料预处理会去掉一些低频词从而控制词典大小，这里我们截取前8000个高频词汇，低频词使用一个统一标识替换（这里是UNKNOWN_TOKEN），在经过预处理之后每一个词得到一个编号；为了学出来哪些词常常作为句子开始和句子结束，引入SENTENCE_START和SENTENCE_END两个特殊字符。具体代码如下：

vocabulary_size = 8000unknown_token = "UNKNOWN_TOKEN"sentence_start_token = "SENTENCE_START"sentence_end_token = "SENTENCE_END" # Read the data and append SENTENCE_START and SENTENCE_END tokensprint "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 wordstokenized_sentences = [nltk.word_tokenize(sent) for sent in sentences] # Count the word frequenciesword_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 vectorsvocab = 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_sizeprint "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 tokenfor 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 dataX_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]

三、网络结构

        循环神经网络的结构如下图：

        RNN网络有状态的概念。如上图，t表示的是状态， xt表示的状态t的输入， st 表示状态t时隐层的输出， ot表示输出。特别的地方在于，隐层的输入有两个来源，一个是当前的 xt输入、一个是上一个状态隐层的输出 st−1。W,U,V 为参数。使用公式可以将上面结构表示为：

      

        参数的初始化有很多种方法，都初始化为0将会导致symmetric calculations ，如何初始化其实是和具体的激活函数有关系，我们这里使用的是tanh，一种推荐的方式是初始化为 [−1/√n,1/√n],其中n是前一层接入的链接数。

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))

四、前向传播

        前向传播代码如下：

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]

        预测函数为：

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)

五、损失函数
        使用交叉熵作为损失函数，如果有N个样本，损失函数可以写为：

        损失函数计算代码：

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)/N

六、BPTT学习参数

        BPTT（Backpropagation Through Time）是一种非常直观的方法，和传统的BP类似，只不过传播的路径是个循环，并且路径上的参数是共享的。损失是交叉熵，损失可以表示为：

        其中 yt是真实值， (̂yt)是预估值，将误差展开可以用图表示为：

        BPTT梯度更新的代码为：

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]

七、梯度弥散现象

        tanh和sigmoid函数和导数的取值返回如下图，可以看到导数取值是[0-1]，用几次链式法则就会将梯度指数级别缩小，所以传播不了几层就会出现梯度非常弱。克服这个问题的LSTM是一种最近比较流行的解决方案。

八、Gradient Checking

         梯度检验是非常有用的，检查的原理是一个点的梯度等于这个点的斜率，估算一个点的斜率可以通过求极限的方式：

        通过比较斜率和梯度的值，我们就可以判断梯度计算的是否有问题。需要注意的是这个检验成本还是很高的，因为我们的参数个数是百万量级的。梯度检验的代码：

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实现

        W=W−λΔW，其中 ΔW就是梯度，具体代码：

# 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
        生成过程其实就是模型的应用过程，只需要反复执行预测函数即可：

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 = 10senten_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)

参考：

①http://www.wildml.com/2015/09/recurrent-neural-networks-tutorial-part-2-implementing-a-language-model-rnn-with-python-numpy-and-theano/