Recurrent Neural Networks are proved to be “Turing Complete” machines. What this means is that they can compute any arbitary function, given infinite resources. This was proved long back (in around 1994) by some neural network researchers. Despite their proof RNN’s were not very popular until recently (for past 4-5 years), it can be attributed to lack of computational resources and lack of engineering efforts in training them (vanishing/exploding gradient problems).
But now it is safe to say that we have virtually infinite resource and infinite data at our disposal. So this property of RNN (Turing Completeness) can be exploited to learn complicated functions. In this post, I’ll demonstrate how RNN can learn to sort an integer sequence with keras using < 100 lines of code. THe model which is used is slight variant of RNN, seq2seq (sequence to sequence model). Seq2seq model is a variation of RNN, where two RNN’s are stacked to learn a many-to-many function (vector function), where input is a sequence and output is also a sequence.
Seq2seq model is an encoder-decoder model where one RNN is used to “encodes” a sequence and other RNN “decodes” the encoded sequence. RNN’s can be seen as compression models which compress sequence information into a fixed length vector (last hidden state of RNN sequence). This fixed length vector “encodes” the input sequence. Now this “compressed” sequence vector is feed into another RNN, which generates output sequence based on information provided by this encoded vector. The whole model is differentiable end-to-end and hence can be trained using traditional Gradient Descent methods (Backprop).
Mathematically sorting can be represented by some complex function (many-to-many) which takes an input an arbitarly arranged vector (sequence) and it generates the sorted vector as output. This function fits into our framework of seq2seq model, so we can train a RNN to learn this function, i.e. we RNN’s can learn to sort.
Ok, enough talk, let’s code !!
Data generation is pretty easy in this case. We can generate random integer sequences (which will serve as input to RNN) and sort these sequences to generate output for RNN. It’s pretty straight-forward in python.
import numpy as np # Neural networks take input as vectors so we have to convert integers to vectors using one-hot encoding # This function will encode a given integer sequence into RNN compatible format (one-hot representation) def encode(X,seq_len, vocab_size): x = np.zeros((len(X),seq_len, vocab_size), dtype=np.float32) for ind,batch in enumerate(X): for j, elem in enumerate(batch): x[ind, j, elem] = 1 return x # This is a generator function which can generate infinite-stream of inputs for training def batch_gen(batch_size=32, seq_len=10, max_no=100): # Randomly generate a batch of integer sequences (X) and its sorted # counterpart (Y) x = np.zeros((batch_size, seq_len, max_no), dtype=np.float32) y = np.zeros((batch_size, seq_len, max_no), dtype=np.float32) while True: # Generates a batch of input X = np.random.randint(max_no, size=(batch_size, seq_len)) Y = np.sort(X, axis=1) for ind,batch in enumerate(X): for j, elem in enumerate(batch): x[ind, j, elem] = 1 for ind,batch in enumerate(Y): for j, elem in enumerate(batch): y[ind, j, elem] = 1 yield x, y x.fill(0.0) y.fill(0.0)
This will generate input batches for RNN. Now lets see how to code-up the seq2seq model
from keras.models import Sequential from keras.layers.core import Activation, RepeatVector, TimeDistributedDense, Dropout, Dense from keras.layers import recurrent import numpy as np from data import batch_gen, encode RNN = recurrent.LSTM # global parameters. batch_size=32 seq_len = 10 max_no = 100 # Initializing model model = Sequential() # This is encoder RNN (we are taking a variant of RNN called LSTM, because plain RNN's suffer from long-term dependencies issues model.add(RNN(100, input_shape=(seq_len, max_no))) # Dropout to enhace RNN's generalization capacities model.add(Dropout(0.25)) # At this point RNN will generate a summary vector of the sequence, so to feed it to decoder we need to repeat it lenght of output seq. number of times model.add(RepeatVector(seq_len)) # Decoder RNN, which will return output sequence model.add(RNN(100, return_sequences=True)) # Adding linear layer at each time step model.add(TimeDistributedDense(max_no)) # Adding non-linearity on top of linear layer at each time-step, since output at each time step is supposed to be probability distribution over max. no of integer in sequence # we add softmax non-linearity model.add(Dropout(0.5)) model.add(Activation('softmax')) # Since this is a multiclass classification task, crossentropy loss is being used. Optimizer is adam, which is a particular instance of adaptive learning rate Gradient Descent methods model.compile(loss='categorical_crossentropy', optimizer='adam', metrics=['accuracy']) # Now the training loop, we'll sample input batches from the generator function written previously and feed it to the RNN for learning for ind,(X,Y) in enumerate(batch_gen(batch_size, seq_len, max_no)): loss, acc = model.train_on_batch(X, Y) # We'll test RNN after each 250 iteration to check how well it is performing if ind % 250 == 0: testX = np.random.randint(max_no, size=(1, seq_len)) test = encode(testX, seq_len, max_no) print testX #pdb.set_trace() y = model.predict(test, batch_size=1) print "actual sorted output is" print np.sort(testX) print "sorting done by RNN is" print np.argmax(y, axis=2) print "\n"
Now the neural network is ready to learn to sort. I’ll post a screenshot of how it learned to sort a relatively short integer sequence.
RNN learning to sort as it goes!! :)