RNN(Recurrent Neural Networks)公式推导和实现

RNN(Recurrent Neural Networks)公式推导和实现

http://x-algo.cn/index.php/2016/04/25/rnn-recurrent-neural-networks-derivation-and-implementation/

2016-04-25 分类：Deep Learning / NLP / RNN 阅读(6997) 评论(7) 

本文主要参考wildml的博客所写，所有的代码都是python实现。没有使用任何深度学习的工具，公式推导虽然枯燥，但是推导一遍之后对RNN的理解会更加的深入。看本文之前建议对传统的神经网络的基本知识已经了解，如果不了解的可以看此文：『神经网络（Neural Network）实现』。

所有可执行代码：Code to follow along is on Github.

文章目录 [展开]

语言模型

熟悉NLP的应该会比较熟悉，就是将自然语言的一句话『概率化』。具体的，如果一个句子有m个词，那么这个句子生成的概率就是：

P(w1,...,wm)=∏mi=1P(wi∣w1,...,wi−1)P(w1,...,wm)=∏i=1mP(wi∣w1,...,wi−1)

其实就是假设下一次词生成的概率和只和句子前面的词有关，例如句子『He went to buy some chocolate』生成的概率可以表示为：  P(他喜欢吃巧克力) = P(他喜欢吃) * P(巧克力|他喜欢吃) 。

数据预处理

训练模型总需要语料，这里语料是来自google big query的reddit的评论数据，语料预处理会去掉一些低频词从而控制词典大小，低频词使用一个统一标识替换（这里是UNKNOWN_TOKEN），预处理之后每一个词都会使用一个唯一的编号替换；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。具体就看代码吧：

点击展开代码

 

网络结构

和传统的nn不同，但是也很好理解，rnn的网络结构如下图：

A recurrent neural network and the unfolding in time of the computation involved in its forward computation.

不同之处就在于rnn是一个『循环网络』，并且有『状态』的概念。

如上图，t表示的是状态， xtxt 表示的状态t的输入， stst 表示状态t时隐层的输出， otot 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 xtxt 输入、一个是上一个状态隐层的输出 st−1st−1 ， W,U,VW,U,V 为参数。使用公式可以将上面结构表示为：

sty^t=tanh(Uxt+Wst−1)=softmax(Vst)st=tanh⁡(Uxt+Wst−1)y^t=softmax(Vst)

如果隐层节点个数为100，字典大小C=8000，参数的维度信息为：

xtotstUVW∈R8000∈R8000∈R100∈R100×8000∈R8000×100∈R100×100xt∈R8000ot∈R8000st∈R100U∈R100×8000V∈R8000×100W∈R100×100

初始化

参数的初始化有很多种方法，都初始化为0将会导致『symmetric calculations 』（我也不懂），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 [−1n√,1n√][−1n,1n] ，其中n是前一层接入的链接数。更多信息请点击查看更多。

 

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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))

前向传播

类似传统的nn的方法，计算几个矩阵乘法即可：

 

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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]

预测函数可以写为：

 

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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)

损失函数

类似nn方法，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：

L(y,o)=−1N∑n∈NynlogonL(y,o)=−1N∑n∈Nynlog⁡on

下面两个函数用来计算损失：

 

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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

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：

Et(yt,y^t)E(y,y^)=−ytlogy^t=∑tEt(yt,y^t)=−∑tytlogy^tEt(yt,y^t)=−ytlog⁡y^tE(y,y^)=∑tEt(yt,y^t)=−∑tytlog⁡y^t

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

所以对所有误差求W的偏导数为：

∂E∂W=∑t∂Et∂W∂E∂W=∑t∂Et∂W

进一步可以将 EtEt 表示为：

∂E3∂V=∂E3∂y^3∂y^3∂V=∂E3∂y^3∂y^3∂z3∂z3∂V=(y^3−y3)⊗s3∂E3∂V=∂E3∂y^3∂y^3∂V=∂E3∂y^3∂y^3∂z3∂z3∂V=(y^3−y3)⊗s3

根据链式法则和RNN中W权值共享，可以得到：

∂E3∂W=∑k=03∂E3∂y^3∂y^3∂s3∂s3∂sk∂sk∂W∂E3∂W=∑k=03∂E3∂y^3∂y^3∂s3∂s3∂sk∂sk∂W

下图将这个过程表示的比较形象

BPTT更新梯度的代码：

 

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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]

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。

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：

∂L∂θ≈limh→0J(θ+h)−J(θ−h)2h∂L∂θ≈limh→0J(θ+h)−J(θ−h)2h

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：

 

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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)

SGD实现

这个公式应该非常熟悉：

W=W−λΔWW=W−λΔW

其中 ΔWΔW 就是梯度，具体代码：

 

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# 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

生成文本

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：

 

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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)

参考文献

Recurrent Neural Networks Tutorial, Part 2 – Implementing a RNN with Python, Numpy and Theano

Recurrent Neural Networks Tutorial, Part 3 – Backpropagation Through Time and Vanishing Gradients