zarzen/logistic_sgd_notes.py

## logistic_sgd_notes.py
# -*- coding: utf-8 -*-
"""
This tutorial introduces logistic regression using Theano and stochastic
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.


References:

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

"""
__docformat__ = 'restructedtext en'

import cPickle
import gzip
import os
import sys
import time

import numpy

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 datapoints lie

        :type n_out: int
        :param n_out: number of output units, the dimension of the space in
                      which the labels lie

        """
        # start-snippet-1
        # 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
        )

        # symbolic expression for computing the matrix of class-membership
        # probabilities
        # Where:
        # W is a matrix where column-k represent the separation hyper plain for
        # class-k
        # x is a matrix where row-j  represents input training sample-j
        # b is a vector where element-k represent the free parameter of hyper
        # plain-k
        self.p_y_given_x = T.nnet.softmax(T.dot(input, self.W) + self.b)

        # symbolic description of how to compute prediction as class whose
        # probability is maximal
        self.y_pred = T.argmax(self.p_y_given_x, axis=1)
        # end-snippet-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
        """
        # start-snippet-2
        # 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])
        # end-snippet-2

    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', y.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
            """
            这个之前一直没看懂，因为无法理解y_pred这个表达可以在不同的minibatch的适合求出
            不同的值,一度认为在创建这个对象时，init方法里，传入了input并且设置了W+b,求出的y_pred应该是个定值。
            后来发现这个还是个符号表达式。这里的关键点在于，下面sgd_optimization_mnist函数中，定义的
            test_model，validate_model函数都使用givens关键字，对于表达式求值时的变量做了重新定义。
            另外一点，295行左右的classifier = LogisticRegression(input=x, n_in=28 * 28, n_out=10)这个，
            传入的input就是那个x，之前没看清楚。这个x是个Tensor,所以那个y_pred后面还是个符号表达式。
            """

            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 data_dir == "" and not os.path.isfile(dataset):
        # Check if dataset is in the data directory.
        new_path = os.path.join(
            os.path.split(__file__)[0],
            "..",
            "data",
            dataset
        )
        if os.path.isfile(new_path) or data_file == 'mnist.pkl.gz':
            dataset = new_path

    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 rval


def sgd_optimization_mnist(learning_rate=0.13, n_epochs=1000,
                           dataset='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

    # generate symbolic variables for input (x and y represent a
    # minibatch)
    x = T.matrix('x')  # data, presented as rasterized images
    y = T.ivector('y')  # labels, 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)

    # start-snippet-3
    # 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]
        }
    )
    # end-snippet-3

    ###############
    # 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_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
            """
            这边使用了early stopping来防止过拟合，但是这个策略不是很懂，就看懂了每个epoch做一个
            validation test，然后如果找到了比之前记录的best表现更好的，就给更多的patience用来训练。

            教程里的early stopping的内容还是能看懂的。大概就是说，用validation test来看，模型在validation
            数据集上的表现，如果表现进步缓慢，甚至是下降了，那么就应该停止训练。然后在实例中，教程使用了几何放大
            的办法来增加patience。
            """
            if (iter + 1) % validation_frequency == 0:#这个判断就是用来判断1个epoch是不是跑完的，我觉得这个写法有点蛋疼啊
                #可能是我没看懂iter的另一个用途。

                # 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)
                        #patience increase就是等于2，就是所谓的几何放大。但是用这么个iteration真的让我有点蛋疼

                    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()
	# -- coding: utf-8 --
	"""
	This tutorial introduces logistic regression using Theano and stochastic
	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.


	References:

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

	"""
	__docformat__ = 'restructedtext en'

	import cPickle
	import gzip
	import os
	import sys
	import time

	import numpy

	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 datapoints lie

	:type n_out: int
	:param n_out: number of output units, the dimension of the space in
	which the labels lie

	"""
	# start-snippet-1
	# 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
	)

	# symbolic expression for computing the matrix of class-membership
	# probabilities
	# Where:
	# W is a matrix where column-k represent the separation hyper plain for
	# class-k
	# x is a matrix where row-j represents input training sample-j
	# b is a vector where element-k represent the free parameter of hyper
	# plain-k
	self.p_y_given_x = T.nnet.softmax(T.dot(input, self.W) + self.b)

	# symbolic description of how to compute prediction as class whose
	# probability is maximal
	self.y_pred = T.argmax(self.p_y_given_x, axis=1)
	# end-snippet-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
	"""
	# start-snippet-2
	# 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])
	# end-snippet-2

	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', y.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
	"""
	这个之前一直没看懂，因为无法理解y_pred这个表达可以在不同的minibatch的适合求出
	不同的值,一度认为在创建这个对象时，init方法里，传入了input并且设置了W+b,求出的y_pred应该是个定值。
	后来发现这个还是个符号表达式。这里的关键点在于，下面sgd_optimization_mnist函数中，定义的
	test_model，validate_model函数都使用givens关键字，对于表达式求值时的变量做了重新定义。
	另外一点，295行左右的classifier = LogisticRegression(input=x, n_in=28 * 28, n_out=10)这个，
	传入的input就是那个x，之前没看清楚。这个x是个Tensor,所以那个y_pred后面还是个符号表达式。
	"""

	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 data_dir == "" and not os.path.isfile(dataset):
	# Check if dataset is in the data directory.
	new_path = os.path.join(
	os.path.split(__file__)[0],
	"..",
	"data",
	dataset
	)
	if os.path.isfile(new_path) or data_file == 'mnist.pkl.gz':
	dataset = new_path

	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 rval


	def sgd_optimization_mnist(learning_rate=0.13, n_epochs=1000,
	dataset='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

	# generate symbolic variables for input (x and y represent a
	# minibatch)
	x = T.matrix('x') # data, presented as rasterized images
	y = T.ivector('y') # labels, 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)

	# start-snippet-3
	# 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]
	}
	)
	# end-snippet-3

	###############
	# 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_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
	"""
	这边使用了early stopping来防止过拟合，但是这个策略不是很懂，就看懂了每个epoch做一个
	validation test，然后如果找到了比之前记录的best表现更好的，就给更多的patience用来训练。

	教程里的early stopping的内容还是能看懂的。大概就是说，用validation test来看，模型在validation
	数据集上的表现，如果表现进步缓慢，甚至是下降了，那么就应该停止训练。然后在实例中，教程使用了几何放大
	的办法来增加patience。
	"""
	if (iter + 1) % validation_frequency == 0:#这个判断就是用来判断1个epoch是不是跑完的，我觉得这个写法有点蛋疼啊
	#可能是我没看懂iter的另一个用途。

	# 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)
	#patience increase就是等于2，就是所谓的几何放大。但是用这么个iteration真的让我有点蛋疼

	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()