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### RNN TUTORIAL, PART 2 – IMPLEMENTING A RNN WITH PYTHON, NUMPY AND THEANO

This the second part of the Recurrent Neural Network Tutorial. The first part is here.

Code to follow along is on Github.

In this part we will implement a full Recurrent Neural Network from scratch using Python and optimize our implementation using Theano, a library to perform operations on a GPU. The full code is available on Github. I will skip over some boilerplate code that is not essential to understanding Recurrent Neural Networks, but all of that is also on Github.

### LANGUAGE MODELING

Our goal is to build a Language Model using a Recurrent Neural Network. Here’s what that means. Let’s say we have sentence of $m$ words. A language model allows us to predict the probability of observing the sentence (in a given dataset) as:

\begin{aligned} P(w_1,...,w_m) = \prod_{i=1}^{m} P(w_i \mid w_1,..., w_{i-1}) \end{aligned}

In words, the probability of a sentence is the product of probabilities of each word given the words that came before it. So, the probability of the sentence “He went to buy some chocolate” would be the probability of “chocolate” given “He went to buy some”, multiplied by the probability of “some” given “He went to buy”, and so on.

Why is that useful? Why would we want to assign a probability to observing a sentence?

First, such a model can be used as a scoring mechanism. For example, a Machine Translation system typically generates multiple candidates for an input sentence. You could use a language model to pick the most probable sentence. Intuitively, the most probable sentence is likely to be grammatically correct. Similar scoring happens in speech recognition systems.

But solving the Language Modeling problem also has a cool side effect. Because we can predict the probability of a word given the preceding words, we are able to generate new text. It’s a generative model. Given an existing sequence of words we sample a next word from the predicted probabilities, and repeat the process until we have a full sentence. Andrej Karparthy has a great post that demonstrates what language models are capable of. His models are trained on single characters as opposed to full words, and can generate anything from Shakespeare to Linux Code.

Note that in the above equation the probability of each word is conditioned on all previous words. In practice, many models have a hard time representing such long-term dependencies due to computational or memory constraints. They are typically limited to looking at only a few of the previous words. RNNs can, in theory, capture such long-term dependencies, but in practice it’s a bit more complex. We’ll explore that in a later post.

### TRAINING DATA AND PREPROCESSING

To train our language model we need text to learn from. Fortunately we don’t need any labels to train a language model, just raw text. I downloaded 15,000 longish reddit comments from a dataset available on Google’s BigQuery. Text generated by our model will sound like reddit commenters (hopefully)! But as with most Machine Learning projects we first need to do some pre-processing to get our data into the right format.

#### 1. TOKENIZE TEXT

We have raw text, but we want to make predictions on a per-word basis. This means we must tokenize our comments into sentences, and sentences into words. We could just split each of the comments by spaces, but that wouldn’t handle punctuation properly. The sentence “He left!” should be 3 tokens: “He”, “left”, “!”. We’ll use NLTK’s word_tokenize and sent_tokenize methods, which do most of the hard work for us.

#### 2. REMOVE INFREQUENT WORDS

Most words in our text will only appear one or two times. It’s a good idea to remove these infrequent words. Having a huge vocabulary will make our model slow to train (we’ll talk about why that is later), and because we don’t have a lot of contextual examples for such words we wouldn’t be able to learn how to use them correctly anyway. That’s quite similar to how humans learn. To really understand how to appropriately use a word you need to have seen it in different contexts.

In our code we limit our vocabulary to the vocabulary_size most common words (which I set to 8000, but feel free to change it). We replace all words not included in our vocabulary by UNKNOWN_TOKEN. For example, if we don’t include the word “nonlinearities” in our vocabulary, the sentence “nonlineraties are important in neural networks” becomes “UNKNOWN_TOKEN are important in Neural Networks”. The word UNKNOWN_TOKEN will become part of our vocabulary and we will predict it just like any other word. When we generate new text we can replace UNKNOWN_TOKEN again, for example by taking a randomly sampled word not in our vocabulary, or we could just generate sentences until we get one that doesn’t contain an unknown token.

#### 3. PREPEND SPECIAL START AND END TOKENS

We also want to learn which words tend start and end a sentence. To do this we prepend a special SENTENCE_START token, and append a special SENTENCE_END token to each sentence. This allows us to ask: Given that the first token is SENTENCE_START, what is the likely next word (the actual first word of the sentence)?

#### 4. BUILD TRAINING DATA MATRICES

The input to our Recurrent Neural Networks are vectors, not strings. So we create a mapping between words and indices, index_to_word, and word_to_index. For example,  the word “friendly” may be at index 2001. A training example $x$ may look like [0, 179, 341, 416], where 0 corresponds to SENTENCE_START. The corresponding label $y$ would be [179, 341, 416, 1]. Remember that our goal is to predict the next word, so y is just the x vector shifted by one position with the last element being the SENTENCE_END token. In other words, the correct prediction for word 179 above would be 341, the actual next word.

 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]

### BUILDING THE RNN

For a general overview of RNNs take a look at first part of the tutorial.

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

Let’s get concrete and see what the RNN for our language model looks like. The input $x$ will be a sequence of words (just like the example printed above) and each $x_t$ is a single word. But there’s one more thing: Because of how matrix multiplication works we can’t simply use a word index (like 36) as an input. Instead, we represent each word as a one-hot vector of size vocabulary_size. For example, the word with index 36 would be the vector of all 0’s and a 1 at position 36. So, each $x_t$ will become a vector, and $x$ will be a matrix, with each row representing a word. We’ll perform this transformation in our Neural Network code instead of doing it in the pre-processing. The output of our network $o$ has a similar format. Each $o_t$ is a vector of vocabulary_size elements, and each element represents the probability of that word being the next word in the sentence.

Let’s recap the equations for the RNN from the first part of the tutorial:

\begin{aligned} s_t &= \tanh(Ux_t + Ws_{t-1}) \\ o_t &= \mathrm{softmax}(Vs_t) \end{aligned}

I always find it useful to write down the dimensions of the matrices and vectors. Let’s assume we pick a vocabulary size $C = 8000$ and a hidden layer size $H = 100$. You can think of the hidden layer size as the “memory” of our network. Making it bigger allows us to learn more complex patterns, but also results in additional computation. Then we have:

\begin{aligned} x_t & \in \mathbb{R}^{8000} \\ o_t & \in \mathbb{R}^{8000} \\ s_t & \in \mathbb{R}^{100} \\ U & \in \mathbb{R}^{100 \times 8000} \\ V & \in \mathbb{R}^{8000 \times 100} \\ W & \in \mathbb{R}^{100 \times 100} \\ \end{aligned}

This is valuable information. Remember that $U,V$ and $W$ are the parameters of our network we want to learn from data. Thus, we need to learn a total of $2HC + H^2$ parameters. In the case of $C=8000$ and $H=100$ that’s 1,610,000. The dimensions also tell us the bottleneck of our model. Note that because $x_t$ is a one-hot vector, multiplying it with $U$ is essentially the same as selecting a column of U, so we don’t need to perform the full multiplication. Then, the biggest matrix multiplication in our network is $Vs_t$. That’s why we want to keep our vocabulary size small if possible.

Armed with this, it’s time to start our implementation.

#### INITIALIZATION

We start by declaring a RNN class an initializing our parameters. I’m calling this class RNNNumpy because we will implement a Theano version later. Initializing the parameters $U,V$ and $W$ is a bit tricky. We can’t just initialize them to 0’s because that would result in symmetric calculations in all our layers. We must initialize them randomly. Because proper initialization seems to have an impact on training results there has been lot of research in this area. It turns out that the best initialization depends on the activation function ($\tanh$ in our case) and one recommended approach is to initialize the weights randomly in the interval from $\left[-\frac{1}{\sqrt{n}}, \frac{1}{\sqrt{n}}\right]$ where $n$ is the number of incoming connections from the previous layer. This may sound overly complicated, but don’t worry too much it. As long as you initialize your parameters to small random values it typically works out fine.

 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))

Above, word_dim is the size of our vocabulary, and hidden_dim is the size of our hidden layer (we can pick it). Don’t worry about the bptt_truncate parameter for now, we’ll explain what that is later.

#### FORWARD PROPAGATION

Next, let’s implement the forward propagation (predicting word probabilities) defined by our equations above:

 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] RNNNumpy.forward_propagation = forward_propagation

We not only return the calculated outputs, but also the hidden states. We will use them later to calculate the gradients, and by returning them here we avoid duplicate computation. Each $o_t$ is a vector of probabilities representing the words in our vocabulary, but sometimes, for example when evaluating our model, all we want is the next word with the highest probability. We call this function predict:

 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) RNNNumpy.predict = predict

Let’s try our newly implemented methods and see an example output:

 np.random.seed(10)model = RNNNumpy(vocabulary_size)o, s = model.forward_propagation(X_train[10])print o.shapeprint o
 (45, 8000)[[ 0.00012408  0.0001244   0.00012603 ...,  0.00012515  0.00012488   0.00012508] [ 0.00012536  0.00012582  0.00012436 ...,  0.00012482  0.00012456   0.00012451] [ 0.00012387  0.0001252   0.00012474 ...,  0.00012559  0.00012588   0.00012551] ...,  [ 0.00012414  0.00012455  0.0001252  ...,  0.00012487  0.00012494   0.0001263 ] [ 0.0001252   0.00012393  0.00012509 ...,  0.00012407  0.00012578   0.00012502] [ 0.00012472  0.0001253   0.00012487 ...,  0.00012463  0.00012536   0.00012665]]

For each word in the sentence (45 above), our model made 8000 predictions representing probabilities of the next word. Note that because we initialized $U,V,W$ to random values these predictions are completely random right now. The following gives the indices of the highest probability predictions for each word:

 predictions = model.predict(X_train[10])print predictions.shapeprint predictions
 (45,)[1284 5221 7653 7430 1013 3562 7366 4860 2212 6601 7299 4556 2481 238 2539 21 6548 261 1780 2005 1810 5376 4146 477 7051 4832 4991 897 3485 21 7291 2007 6006 760 4864 2182 6569 2800 2752 6821 4437 7021 7875 6912 3575]

#### CALCULATING THE LOSS

To train our network we need a way to measure the errors it makes. We call this the loss function $L$, and our goal is find the parameters $U,V$ and $W$ that minimize the loss function for our training data. A common choice for the loss function is the cross-entropy loss. If we have $N$ training examples (words in our text) and $C$ classes (the size of our vocabulary) then the loss with respect to our predictions $o$ and the true labels $y$ is given by:

\begin{aligned} L(y,o) = - \frac{1}{N} \sum_{n \in N} y_{n} \log o_{n} \end{aligned}

The formula looks a bit complicated, but all it really does is sum over our training examples and add to the loss based on how off our prediction are. The further away $y$ (the correct words) and $o$ (our predictions), the greater the loss will be. We implement the function calculate_loss:

 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 RNNNumpy.calculate_total_loss = calculate_total_lossRNNNumpy.calculate_loss = calculate_loss

Let’s take a step back and think about what the loss should be for random predictions. That will give us a baseline and make sure our implementation is correct. We have $C$ words in our vocabulary, so each word should be (on average) predicted with probability $1/C$, which would yield a loss of $L = -\frac{1}{N} N \log\frac{1}{C} = \log C$:

 # Limit to 1000 examples to save timeprint "Expected Loss for random predictions: %f" % np.log(vocabulary_size)print "Actual loss: %f" % model.calculate_loss(X_train[:1000], y_train[:1000])
 Expected Loss for random predictions: 8.987197Actual loss: 8.987440

Pretty close! Keep in mind that evaluating the loss on the full dataset is an expensive operation and can take hours if you have a lot of data!

#### TRAINING THE RNN WITH SGD AND BACKPROPAGATION THROUGH TIME (BPTT)

Remember that we want to find the parameters $U,V$ and $W$ that minimize the total loss on the training data. The most common way to do this is SGD, Stochastic Gradient Descent. The idea behind SGD is pretty simple. We iterate over all our training examples and during each iteration we nudge the parameters into a direction that reduces the error. These directions are given by the gradients on the loss: $\frac{\partial L}{\partial U}, \frac{\partial L}{\partial V}, \frac{\partial L}{\partial W}$. SGD also needs a learning rate, which defines how big of a step we want to make in each iteration. SGD is the most popular optimization method not only for Neural Networks, but also for many other Machine Learning algorithms. As such there has been a lot of research on how to optimize SGD using batching, parallelism and adaptive learning rates. Even though the basic idea is simple, implementing SGD in a really efficient way can become very complex. If you want to learn more about SGD this is a good place to start. Due to its popularity there are a wealth of tutorials floating around the web, and I don’t want to duplicate them here. I’ll implement a simple version of SGD that should be understandable even without a background in optimization.

But how do we calculate those gradients we mentioned above? In a traditional Neural Network we do this through the backpropagation algorithm. In RNNs we use a slightly modified version of the this algorithm called Backpropagation Through Time (BPTT). Because the parameters are shared by all time steps in the network, the gradient at each output depends not only on the calculations of the current time step, but also the previous time steps. If you know calculus, it really is just applying the chain rule. The next part of the tutorial will be all about BPTT, so I won’t go into detailed derivation here. For a general introduction to backpropagation check out this and this post. For now you can treat BPTT as a black box. It takes as input a training example $(x,y)$ and returns the gradients $\frac{\partial L}{\partial U}, \frac{\partial L}{\partial V}, \frac{\partial L}{\partial W}$.

 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        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)            dLdW += np.outer(delta_t, s[bptt_step-1])                          dLdU[:,x[bptt_step]] += delta_t            # Update delta for next step            delta_t = self.W.T.dot(delta_t) * (1 - s[bptt_step-1] ** 2)    return [dLdU, dLdV, dLdW] RNNNumpy.bptt = bptt

\begin{aligned} \frac{\partial L}{\partial \theta} \approx \lim_{h \to 0} \frac{J(\theta + h) - J(\theta -h)}{2h} \end{aligned}
 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) RNNNumpy.gradient_check = gradient_check # To avoid performing millions of expensive calculations we use a smaller vocabulary size for checking.grad_check_vocab_size = 100np.random.seed(10)model = RNNNumpy(grad_check_vocab_size, 10, bptt_truncate=1000)model.gradient_check([0,1,2,3], [1,2,3,4])