LogisticRegression.py 解析

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"""This tutorial introduces logistic regression using Theano and stochasticgradient descent.Logistic regression is a probabilistic, linear classifier. It is parametrizedby a weight matrix :math:`W` and a bias vector :math:`b`. Classification isdone by projecting data points onto a set of hyperplanes, the distance towhich 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 ofthe 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 methodsuitable for large datasets, and a conjugate gradient optimization methodthat is suitable for smaller datasets.References:    - textbooks: "Pattern Recognition and Machine Learning" -                 Christopher M. Bishop, section 4.3.2"""__docformat__ = 'restructedtext en'import cPickleimport gzipimport osimport sysimport timeimport numpyimport theanoimport theano.tensor as Tclass 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 datapoints lie        :type n_out: int        :param n_out: number of output units, the dimension of the space in                      which the labels lie        """        # initialize with 0 the weights W as a matrix of shape (n_in, n_out)        self.W = theano.shared(value=numpy.zeros((n_in, n_out),                                                 dtype=theano.config.floatX),                                name='W', borrow=True)        # initialize the baises b as a vector of n_out 0s        self.b = theano.shared(value=numpy.zeros((n_out,),                                                 dtype=theano.config.floatX),                               name='b', borrow=True)        # 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)        # parameters of the model        self.params = [self.W, self.b]    def negative_log_likelihood(self, y):        """Return the mean of 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        Note: we use the mean instead of the sum so that              the learning rate is less dependent on the batch size        """        # y.shape[0] is (symbolically) the number of rows in y, i.e.,        # number of examples (call it n) in the minibatch        # T.arange(y.shape[0]) is a symbolic vector which will contain        # [0,1,2,... n-1] T.log(self.p_y_given_x) is a matrix of        # Log-Probabilities (call it LP) with one row per example and        # one column per class LP[T.arange(y.shape[0]),y] is a vector        # v containing [LP[0,y[0]], LP[1,y[1]], LP[2,y[2]], ...,        # LP[n-1,y[n-1]]] and T.mean(LP[T.arange(y.shape[0]),y]) is        # the mean (across minibatch examples) of the elements in v,        # i.e., the mean log-likelihood across the minibatch.        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 ; zero one        loss over the size 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 load_data(dataset):    ''' Loads the dataset    :type dataset: string    :param dataset: the path to the dataset (here MNIST)    '''    #############    # LOAD DATA #    #############    # Download the MNIST dataset if it is not present    data_dir, data_file = os.path.split(dataset)    if (not os.path.isfile(dataset)) and data_file == 'mnist.pkl.gz':        import urllib        origin = 'http://www.iro.umontreal.ca/~lisa/deep/data/mnist/mnist.pkl.gz'        print 'Downloading data from %s' % origin        urllib.urlretrieve(origin, dataset)    print '... loading data'    # Load the dataset    f = gzip.open(dataset, 'rb')    train_set, valid_set, test_set = cPickle.load(f)    f.close()    #train_set, valid_set, test_set format: tuple(input, target)    #input is an numpy.ndarray of 2 dimensions (a matrix)    #witch row's correspond to an example. target is a    #numpy.ndarray of 1 dimensions (vector)) that have the same length as    #the number of rows in the input. It should give the target    #target to the example with the same index in the input.    def shared_dataset(data_xy, borrow=True):        """ 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),                                 borrow=borrow)        shared_y = theano.shared(numpy.asarray(data_y,                                               dtype=theano.config.floatX),                                 borrow=borrow)        # 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)    rval = [(train_set_x, train_set_y), (valid_set_x, valid_set_y),            (test_set_x, test_set_y)]    return rvaldef sgd_optimization_mnist(learning_rate=0.13, n_epochs=1000,                           dataset='../data/mnist.pkl.gz',                           batch_size=600):    """    Demonstrate stochastic gradient descent optimization of a log-linear    model    This is demonstrated on MNIST.    :type learning_rate: float    :param learning_rate: learning rate used (factor for the stochastic                          gradient)    :type n_epochs: int    :param n_epochs: maximal number of epochs to run the optimizer    :type dataset: string    :param dataset: the path of the MNIST dataset file from                 http://www.iro.umontreal.ca/~lisa/deep/data/mnist/mnist.pkl.gz    """    datasets = load_data(dataset)    train_set_x, train_set_y = datasets[0]    valid_set_x, valid_set_y = datasets[1]    test_set_x, test_set_y = datasets[2]    # compute number of minibatches for training, validation and testing    n_train_batches = train_set_x.get_value(borrow=True).shape[0] / batch_size    n_valid_batches = valid_set_x.get_value(borrow=True).shape[0] / batch_size    n_test_batches = test_set_x.get_value(borrow=True).shape[0] / batch_size    ######################    # BUILD ACTUAL MODEL #    ######################    print '... building the model'    # allocate symbolic variables for the data    index = T.lscalar()  # index to a [mini]batch    x = T.matrix('x')  # the data is presented as rasterized images    y = T.ivector('y')  # the labels are presented as 1D vector of                           # [int] labels    # construct the logistic regression class    # Each MNIST image has size 28*28    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)    # compiling a Theano function that computes the mistakes that are made by    # the model on a minibatch    test_model = theano.function(inputs=[index],            outputs=classifier.errors(y),            givens={                x: test_set_x[index * batch_size: (index + 1) * batch_size],                y: test_set_y[index * batch_size: (index + 1) * batch_size]})    validate_model = theano.function(inputs=[index],            outputs=classifier.errors(y),            givens={                x: valid_set_x[index * batch_size:(index + 1) * batch_size],                y: valid_set_y[index * batch_size:(index + 1) * batch_size]})    # compute the gradient of cost with respect to theta = (W,b)    g_W = T.grad(cost=cost, wrt=classifier.W)    g_b = T.grad(cost=cost, wrt=classifier.b)    # specify how to update the parameters of the model as a list of    # (variable, update expression) pairs.    updates = [(classifier.W, classifier.W - learning_rate * g_W),               (classifier.b, classifier.b - learning_rate * g_b)]    # compiling a Theano function `train_model` that returns the cost, but in    # the same time updates the parameter of the model based on the rules    # defined in `updates`    train_model = theano.function(inputs=[index],            outputs=cost,            updates=updates,            givens={                x: train_set_x[index * batch_size:(index + 1) * batch_size],                y: train_set_y[index * batch_size:(index + 1) * batch_size]})    ###############    # TRAIN MODEL #    ###############    print '... training the model'    # early-stopping parameters    patience = 5000  # look as this many examples regardless    patience_increase = 2  # wait this much longer when a new best is                                  # found    improvement_threshold = 0.995  # a relative improvement of this much is                                  # considered significant    validation_frequency = min(n_train_batches, patience / 2)                                  # go through this many                                  # minibatche before checking the network                                  # on the validation set; in this case we                                  # check every epoch    best_params = None    best_validation_loss = numpy.inf    test_score = 0.    start_time = time.clock()    done_looping = False    epoch = 0    while (epoch < n_epochs) and (not done_looping):        epoch = epoch + 1        for minibatch_index in xrange(n_train_batches):            minibatch_avg_cost = train_model(minibatch_index)            # iteration number            iter = (epoch - 1) * n_train_batches + minibatch_index            if (iter + 1) % validation_frequency == 0:                # compute zero-one loss on validation set                validation_losses = [validate_model(i)                                     for i in xrange(n_valid_batches)]                this_validation_loss = numpy.mean(validation_losses)                print('epoch %i, minibatch %i/%i, validation error %f %%' % \                    (epoch, minibatch_index + 1, n_train_batches,                    this_validation_loss * 100.))                # if we got the best validation score until now                if this_validation_loss < best_validation_loss:                    #improve patience if loss improvement is good enough                    if this_validation_loss < best_validation_loss *  \                       improvement_threshold:                        patience = max(patience, iter * patience_increase)                    best_validation_loss = this_validation_loss                    # test it on the test set                    test_losses = [test_model(i)                                   for i in xrange(n_test_batches)]                    test_score = numpy.mean(test_losses)                    print(('     epoch %i, minibatch %i/%i, test error of best'                       ' model %f %%') %                        (epoch, minibatch_index + 1, n_train_batches,                         test_score * 100.))            if patience <= iter:                done_looping = True                break    end_time = time.clock()    print(('Optimization complete with best validation score of %f %%,'           'with test performance %f %%') %                 (best_validation_loss * 100., test_score * 100.))    print 'The code run for %d epochs, with %f epochs/sec' % (        epoch, 1. * epoch / (end_time - start_time))    print >> sys.stderr, ('The code for file ' +                          os.path.split(__file__)[1] +                          ' ran for %.1fs' % ((end_time - start_time)))if __name__ == '__main__':    sgd_optimization_mnist()

ps：其中datasets是

中的变量值，是minist数据库中的信息，gradient descent部分是难点，应该参考http://blog.csdn.net/woshiduan/article/details/9257191 的gradient descent部分进行学习