simple_rnn.ipynb

Kernel: Python 3

Recurrent Neural Networks in Theano

Credits: Forked from summerschool2015 by mila-udem

First, we import some dependencies:

%matplotlib inline
from synthetic import mackey_glass
import matplotlib.pyplot as plt
import theano
import theano.tensor as T
import numpy

floatX = theano.config.floatX

We now define a class that uses scan to initialize an RNN and apply it to a sequence of data vectors. The constructor initializes the shared variables after which the instance can be called on a symbolic variable to construct an RNN graph. Note that this class only handles the computation of the hidden layer activations. We'll define a set of output weights later.

class SimpleRNN(object):
    def __init__(self, input_dim, recurrent_dim):
        w_xh = numpy.random.normal(0, .01, (input_dim, recurrent_dim))
        w_hh = numpy.random.normal(0, .02, (recurrent_dim, recurrent_dim))
        self.w_xh = theano.shared(numpy.asarray(w_xh, dtype=floatX), name='w_xh')
        self.w_hh = theano.shared(numpy.asarray(w_hh, dtype=floatX), name='w_hh')
        self.b_h = theano.shared(numpy.zeros((recurrent_dim,), dtype=floatX), name='b_h')
        self.parameters = [self.w_xh, self.w_hh, self.b_h]
        
    def _step(self, input_t, previous):
            return T.tanh(T.dot(previous, self.w_hh) + input_t)
        
    def __call__(self, x):
        x_w_xh = T.dot(x, self.w_xh) + self.b_h      
        result, updates = theano.scan(self._step,
                                      sequences=[x_w_xh],
                                      outputs_info=[T.zeros_like(self.b_h)])
        return result

For visualization purposes and to keep the optimization time managable, we will train the RNN on a short synthetic chaotic time series. Let's first have a look at the data:

data = numpy.asarray(mackey_glass(2000)[0], dtype=floatX)
plt.plot(data)
plt.show()
data_train = data[:1500]
data_val = data[1500:]

To train an RNN model on this sequences, we need to generate a theano graph that computes the cost and its gradient. In this case, the task will be to predict the next time step and the error objective will be the mean squared error (MSE). We also need to define shared variables for the output weights. Finally, we also add a regularization term to the cost.

w_ho_np = numpy.random.normal(0, .01, (15, 1))
w_ho = theano.shared(numpy.asarray(w_ho_np, dtype=floatX), name='w_ho')
b_o = theano.shared(numpy.zeros((1,), dtype=floatX), name='b_o')

x = T.matrix('x')
my_rnn = SimpleRNN(1, 15)
hidden = my_rnn(x)
prediction = T.dot(hidden, w_ho) + b_o
parameters = my_rnn.parameters + [w_ho, b_o]
l2 = sum((p**2).sum() for p in parameters)
mse = T.mean((prediction[:-1] - x[1:])**2)
cost = mse + .0001 * l2
gradient = T.grad(cost, wrt=parameters)

We now compile the function that will update the parameters of the model using gradient descent.

lr = .3
updates = [(par, par - lr * gra) for par, gra in zip(parameters, gradient)] 
update_model = theano.function([x], cost, updates=updates)
get_cost = theano.function([x], mse)
predict = theano.function([x], prediction)
get_hidden = theano.function([x], hidden)
get_gradient = theano.function([x], gradient)

We can now train the network by supplying this function with our data and calling it repeatedly.

for i in range(1001):
    mse_train = update_model(data_train)
    
    if i % 100 == 0:
        mse_val = get_cost(data_val)
        print 'Epoch {}: train mse: {}    validation mse: {}'.format(i, mse_train, mse_val)

Epoch 0: train mse: 0.0515556260943    validation mse: 0.0469637289643
Epoch 100: train mse: 0.0407442860305    validation mse: 0.0401079840958
Epoch 200: train mse: 0.00225670542568    validation mse: 0.00203324528411
Epoch 300: train mse: 0.00185390282422    validation mse: 0.00163305236492
Epoch 400: train mse: 0.00161687470973    validation mse: 0.00139373319689
Epoch 500: train mse: 0.00145859015174    validation mse: 0.00123134546448
Epoch 600: train mse: 0.00134439510293    validation mse: 0.00111229927279
Epoch 700: train mse: 0.00125755299814    validation mse: 0.00102029775735
Epoch 800: train mse: 0.0011889107991    validation mse: 0.000946390733588
Epoch 900: train mse: 0.00113300536759    validation mse: 0.000885214598384
Epoch 1000: train mse: 0.00108635553624    validation mse: 0.000833337020595

Since we're only looking at a very small toy problem here, the model probably already memorized the train data quite well. Let's find out by plotting the predictions of the network:

predict = theano.function([x], prediction)
prediction_np = predict(data)
plt.plot(data[1:], label='data')
plt.plot(prediction_np, label='prediction')
plt.legend()
plt.show()

Small scale optimizations of this type often benefit from more advanced second order methods. The following block defines some functions that allow you to experiment with off-the-shelf optimization routines. In this case we used BFGS.

def vector_to_params(v):
    return_list = []
    offset = 0
    # note the global variable here
    for par in parameters:
        par_size = numpy.product(par.get_value().shape)
        return_list.append(v[offset:offset+par_size].reshape(par.get_value().shape))
        offset += par_size
    return return_list
       
    
def set_params(values):
    for parameter, value in zip(parameters, values):
        parameter.set_value(numpy.asarray(value, dtype=floatX))
        
        
def f_obj(x):
    values = vector_to_params(x)
    set_params(values)
    return get_cost(data_train)
        
    
def f_prime(x):
    values = vector_to_params(x)
    set_params(values)
    grad = get_gradient(data_train)
    return numpy.asarray(numpy.concatenate([var.flatten() for var in grad]), dtype='float64')
    
    
from scipy.optimize import fmin_bfgs
x0 = numpy.asarray(numpy.concatenate([p.get_value().flatten() for p in parameters]), dtype='float64')
result = fmin_bfgs(f_obj, x0, f_prime)

print 'train mse: {}   validation mse: {}'.format(get_cost(data_train), get_cost(data_val))

Warning: Desired error not necessarily achieved due to precision loss.
         Current function value: 0.000218
         Iterations: 5
         Function evaluations: 31
         Gradient evaluations: 19
train mse: 0.000217512235395   validation mse: 0.00018158860621

Generating sequences

Predicting a single step ahead is a relatively easy task. It would be more intresting to see if the network actually learned how to generate multiple time steps such that it can continue the sequence. Write code that generates the next 1000 examples after processing the train sequence.

x_t = T.vector()
h_p = T.vector()
preactivation = T.dot(x_t, my_rnn.w_xh) + my_rnn.b_h
h_t = my_rnn._step(preactivation, h_p)
o_t = T.dot(h_t, w_ho) + b_o

single_step = theano.function([x_t, h_p], [o_t, h_t])

def generate(single_step, x_t, h_p, n_steps):
    output = numpy.zeros((n_steps, 1))
    for output_t in output:
        x_t, h_p = single_step(x_t, h_p)
        output_t[:] = x_t
    return output


output = predict(data_train)
hidden = get_hidden(data_train)

output = generate(single_step, output[-1], hidden[-1], n_steps=200)
plt.plot(output)
plt.plot(data_val[:200])
plt.show()

#Things to Try The quality of the generated sequence is probably not very good. Let's try to improve on it. Things to consider are:

• The initial weight values

• Using L2/L1 regularization

• Using weight noise

• The number of hidden units

• The non-linearity

• Adding direct connections between the input and the output