"""
This tutorial introduces logistic regression using Theano and conjugate 
gradient descent.  

Logistic regression is a probabilistic, linear classifier. It is parametrized
by a weight matrix :math:`W` and a bias vector :math:`b`. Classification is
done by projecting data points onto a set of hyperplanes, the distance to
which is used to determine a class membership probability. 

Mathematically, this can be written as:

.. math::
  P(Y=i|x, W,b) &= softmax_i(W x + b) \\
                &= \frac {e^{W_i x + b_i}} {\sum_j e^{W_j x + b_j}}


The output of the model or prediction is then done by taking the argmax of 
the vector whose i'th element is P(Y=i|x).

.. math::

  y_{pred} = argmax_i P(Y=i|x,W,b)


This tutorial presents a stochastic gradient descent optimization method 
suitable for large datasets, and a conjugate gradient optimization method 
that is suitable for smaller datasets.


References:

   - textbooks: "Pattern Recognition and Machine Learning" - 
                 Christopher M. Bishop, section 4.3.2


"""
__docformat__ = 'restructedtext en'


import numpy, time, cPickle, gzip

import theano
import theano.tensor as T


class LogisticRegression(object):
    """Multi-class Logistic Regression Class

    The logistic regression is fully described by a weight matrix :math:`W` 
    and bias vector :math:`b`. Classification is done by projecting data 
    points onto a set of hyperplanes, the distance to which is used to 
    determine a class membership probability. 
    """




    def __init__(self, input, n_in, n_out):
        """ Initialize the parameters of the logistic regression

        :type input: theano.tensor.TensorType
        :param input: symbolic variable that describes the input of the 
                      architecture ( one minibatch)

        :type n_in: int
        :param n_in: number of input units, the dimension of the space in 
                     which the datapoint lies

        :type n_out: int
        :param n_out: number of output units, the dimension of the space in 
                      which the target lies

        """ 

        # initialize theta = (W,b) with 0s; W gets the shape (n_in, n_out), 
        # while b is a vector of n_out elements, making theta a vector of
        # n_in*n_out + n_out elements
        self.theta = theano.shared( value = numpy.zeros(n_in*n_out+n_out, dtype = theano.config.floatX) )
        # W is represented by the fisr n_in*n_out elements of theta
        self.W     = self.theta[0:n_in*n_out].reshape((n_in,n_out))
        # b is the rest (last n_out elements)
        self.b     = self.theta[n_in*n_out:n_in*n_out+n_out]


        # compute vector of class-membership probabilities in symbolic form
        self.p_y_given_x = T.nnet.softmax(T.dot(input, self.W)+self.b)

        # compute prediction as class whose probability is maximal in 
        # symbolic form
        self.y_pred=T.argmax(self.p_y_given_x, axis=1)





    def negative_log_likelihood(self, y):
        """Return the negative log-likelihood of the prediction of this model
        under a given target distribution.  

        .. math::

            \frac{1}{|\mathcal{D}|}\mathcal{L} (\theta=\{W,b\}, \mathcal{D}) = 
            \frac{1}{|\mathcal{D}|}\sum_{i=0}^{|\mathcal{D}|} \log(P(Y=y^{(i)}|x^{(i)}, W,b)) \\
                \ell (\theta=\{W,b\}, \mathcal{D}) 

        :type y: theano.tensor.TensorType
        :param y: corresponds to a vector that gives for each example the
                  correct label
        """
        return -T.mean(T.log(self.p_y_given_x)[T.arange(y.shape[0]),y])





    def errors(self, y):
        """Return a float representing the number of errors in the minibatch 
        over the total number of examples of the minibatch 

        :type y: theano.tensor.TensorType
        :param y: corresponds to a vector that gives for each example
                  the correct label
        """

        # check if y has same dimension of y_pred 
        if y.ndim != self.y_pred.ndim:
            raise TypeError('y should have the same shape as self.y_pred', 
                ('y', target.type, 'y_pred', self.y_pred.type))
        # check if y is of the correct datatype        
        if y.dtype.startswith('int'):
            # the T.neq operator returns a vector of 0s and 1s, where 1
            # represents a mistake in prediction
            return T.mean(T.neq(self.y_pred, y))
        else:
            raise NotImplementedError()







def cg_optimization_mnist( n_epochs=50, mnist_pkl_gz='mnist.pkl.gz' ):
    """Demonstrate conjugate gradient optimization of a log-linear model 

    This is demonstrated on MNIST.
    
    :type n_epochs: int
    :param n_epochs: number of epochs to run the optimizer 

    :type mnist_pkl_gz: string
    :param mnist_pkl_gz: the path of the mnist training file from 
                         http://www.iro.umontreal.ca/~lisa/deep/data/mnist/mnist.pkl.gz

    """
    #############
    # LOAD DATA #
    #############
    print '... loading data'

    # Load the dataset 
    f = gzip.open(mnist_pkl_gz,'rb')
    train_set, valid_set, test_set = cPickle.load(f)
    f.close()

    def shared_dataset(data_xy):
        """ Function that loads the dataset into shared variables
        
        The reason we store our dataset in shared variables is to allow 
        Theano to copy it into the GPU memory (when code is run on GPU). 
        Since copying data into the GPU is slow, copying a minibatch everytime
        is needed (the default behaviour if the data is not in a shared 
        variable) would lead to a large decrease in performance.
        """
        data_x, data_y = data_xy
        shared_x = theano.shared(numpy.asarray(data_x, dtype=theano.config.floatX))
        shared_y = theano.shared(numpy.asarray(data_y, dtype=theano.config.floatX))
        # When storing data on the GPU it has to be stored as floats
        # therefore we will store the labels as ``floatX`` as well
        # (``shared_y`` does exactly that). But during our computations
        # we need them as ints (we use labels as index, and if they are 
        # floats it doesn't make sense) therefore instead of returning 
        # ``shared_y`` we will have to cast it to int. This little hack
        # lets ous get around this issue
        return shared_x, T.cast(shared_y, 'int32')


    test_set_x,  test_set_y  = shared_dataset(test_set)
    valid_set_x, valid_set_y = shared_dataset(valid_set)
    train_set_x, train_set_y = shared_dataset(train_set)

    batch_size = 600    # size of the minibatch

    n_train_batches = train_set_x.value.shape[0] / batch_size
    n_valid_batches = valid_set_x.value.shape[0] / batch_size
    n_test_batches  = test_set_x.value.shape[0]  / batch_size


    ishape     = (28,28) # this is the size of MNIST images
    n_in       = 28*28   # number of input units
    n_out      = 10      # number of output units


    ######################
    # BUILD ACTUAL MODEL #
    ###################### 
    print '... building the model'

    # allocate symbolic variables for the data
    minibatch_offset = T.lscalar() # offset to the start of a [mini]batch 
    x = T.matrix()   # the data is presented as rasterized images
    y = T.ivector()  # the labels are presented as 1D vector of 
                     # [int] labels

 
    # construct the logistic regression class
    classifier = LogisticRegression( input=x, n_in=28*28, n_out=10)

    # the cost we minimize during training is the negative log likelihood of 
    # the model in symbolic format
    cost = classifier.negative_log_likelihood(y).mean() 

    # compile a theano function that computes the mistakes that are made by 
    # the model on a minibatch
    test_model = theano.function([minibatch_offset], classifier.errors(y),
            givens={
                x:test_set_x[minibatch_offset:minibatch_offset+batch_size],
                y:test_set_y[minibatch_offset:minibatch_offset+batch_size]})

    validate_model = theano.function([minibatch_offset],classifier.errors(y),
            givens={
                x:valid_set_x[minibatch_offset:minibatch_offset+batch_size],
                y:valid_set_y[minibatch_offset:minibatch_offset+batch_size]})

    #  compile a thenao function that returns the cost of a minibatch 
    batch_cost = theano.function([minibatch_offset], cost, 
            givens= {
                x : train_set_x[minibatch_offset:minibatch_offset+batch_size],
                y : train_set_y[minibatch_offset:minibatch_offset+batch_size]})


    
    # compile a theano function that returns the gradient of the minibatch 
    # with respect to theta
    batch_grad = theano.function([minibatch_offset], T.grad(cost,classifier.theta), 
            givens= {
                x : train_set_x[minibatch_offset:minibatch_offset+batch_size],
                y : train_set_y[minibatch_offset:minibatch_offset+batch_size]})


    # creates a function that computes the average cost on the training set
    def train_fn(theta_value):
        classifier.theta.value = theta_value
        train_losses = [batch_cost(i*batch_size) for i in xrange(n_train_batches)]
        return numpy.mean(train_losses)

    # creates a function that computes the average gradient of cost with 
    # respect to theta
    def train_fn_grad(theta_value):
        classifier.theta.value = theta_value
        grad = batch_grad(0)
        for i in xrange(1,n_train_batches):
            grad += batch_grad(i*batch_size)
        return grad/n_train_batches


    validation_scores = [float('inf'), 0]
 
    # creates the validation function
    def callback(theta_value):
        classifier.theta.value = theta_value
        #compute the validation loss
        validation_losses = [validate_model(i*batch_size) for i in xrange(n_valid_batches)]
        this_validation_loss = numpy.mean(validation_losses)
        print('validation error %f %%' % (this_validation_loss*100.,))
        
        # check if it is better then best validation score got until now
        if this_validation_loss < validation_scores[0]:
            # if so, replace the old one, and compute the score on the 
            # testing dataset
            validation_scores[0] = this_validation_loss
            test_loses = [test_model(i*batch_size) for i in xrange(n_test_batches)]
            validation_scores[1] = numpy.mean(test_loses)

    ###############
    # TRAIN MODEL #
    ###############
 
    # using scipy conjugate gradient optimizer 
    import scipy.optimize
    print ("Optimizing using scipy.optimize.fmin_cg...")
    start_time = time.clock()
    best_w_b = scipy.optimize.fmin_cg(
               f        = train_fn, 
               x0       = numpy.zeros((n_in+1)*n_out, dtype=x.dtype),
               fprime   = train_fn_grad,
               callback = callback,
               disp     = 0,
               maxiter  = n_epochs)
    end_time = time.clock()
    print(('Optimization complete with best validation score of %f %%, with '
          'test performance %f %%') % 
               (validation_scores[0]*100., validation_scores[1]*100.))

    print ('The code ran for %f minutes' % ((end_time-start_time)/60.))


if __name__ == '__main__':
    cg_optimization_mnist()