使用逻辑回归进行MNIST分类（Classifying MNIST using Logistic Regressing）

本节假定读者属性了下面的Theano概念：共享变量（shared variable）, 基本数学算子（basic arithmetic ops）, Theano的进阶（T.grad）, floatX(默认为float64)。假如你想要在你的GPU上跑你的代码，你也需要看GPU。

本节的所有代码可以在这里下载。

在这一节，我们将展示Theano如何实现最基本的分类器：逻辑回归分类器。我们以模型的快速入门开始，复习(refresher)和巩固(anchor)数学负号，也展示了数学表达式如何映射到Theano图中。

模型

逻辑回归模型是一个线性概率模型。它由一个权值矩阵W和偏置向量b参数化。分类通过将输入向量提交到一组超平面，每个超平面对应一个类。输入向量和超平面的距离是这个输入属于该类的一个概率量化。

在给定模型下，输入x，输出为y的概率，可以用如下公式表示





Theano代码如下。

# 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)

由于模型的参数需要不断的存取和修正，所以我们把W和b定义为共享变量。这个dot（点乘）和softmax运算用以计算这个P(Y|x,W,b)。这个结果

p_y_given_x

(probability)是一个vector类型的概率向量。

为了获得实际的模型预测，我们使用

T_argmax

操作，来返回

p_y_given_x

的最大值对应的y。

如果想要获得完整的Theano算子，看算子列表

定义一个损失函数

学习优化模型参数需要最小化一个损失参数。在多分类的逻辑回归中，很显然是使用负对数似然函数作为损失函数。似然函数和损失函数定义如下：



虽然整本书都致力于探讨最小化话题，但梯度下降是迄今为止最简单的最小化非线性函数的方法。在这个教程中，我们使用minibatch随机梯度下降算法。可以看随机梯度下降来获得更多细节。

下面的代码定义了一个对给定的minibatch的损失函数。

# 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])

在这里我们使用错误的平均来表示损失函数，以减少minibatch尺寸对我们的影响。

创建一个逻辑回归类

现在，我们要定义一个

逻辑回归

的类，来概括逻辑回归的基本行为。代码已经是我们之前涵盖的了，不再进行过多解释。

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
return T.mean(T.neq(self.y_pred, y))
else:
raise NotImplementedError()

我们通过如下代码来实例化这个类。

# 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)

需要注意的是，输入向量x，和其相关的标签y都是定义在

LogisticRegression

实体外的。这个类需要将输入数据作为

__init__

函数的参数。这在将这些类的实例连接起来构建深网络方面非常有用。一层的输出可以作为下一层的输入。

最后，我们定义了一个

cost

变量来最小化。

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

学习模型

在实现MSGD的许多语言中，需要通过手动求解损失函数对每个参数的梯度（微分）来实现。

在Theano中呢，这是非常简单的。它自动微分，并且使用了一定的数学转换来提高数学稳定性。

g_W = T.grad(cost=cost, wrt=classifier.W)
g_b = T.grad(cost=cost, wrt=classifier.b)

这个函数

train_model

可以被定义如下。

# 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]
}
)

update

是一个list，用以更新每一步的参数。

given

是一个字典，用以表示象征变量，和你在该步中表示的数据。这个

train_model

定义如下：

* 输入是minibatch的

index

，batch的大小之前已经固定，以此被定义为x，以及其相关的y。

* 返回是该

index

下与x，y相关的

cost

/损失函数。

* 每一次函数调用，它都先用index对应的训练集的切片来更新x，y。然后计算该minibatch下的cost，以及申请

update

操作。

每次

train_model(inedx)

被调用，它都计算并返回该minibatch的cost，当然这也是MSGD的一步。整个学习算法因循环了数据集所有样例。

训练模型

在之前论述中所说，我们对分类错误的样本感兴趣（不仅仅是可能性）。因此模型中增加了一个额外的实例方法，来纪录每个minibatch中的错误分类样例数。

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
return T.mean(T.neq(self.y_pred, y))
else:
raise NotImplementedError()

我们创建了

test_model

函数，然后也创建了

validate_model

来调用去修正这个值。当然

validate_model

是early-stopping的关键。它们都是来统计minibatch中分类错误的样例数。

# 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]
}
)

把它们组合起来

最后的代码如下。

"""
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
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

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()

这个输出将是如下的格式



在

Intel® Core™ i3-2100 CPU @ 3.10GHz × 4

上，这个代码的速度是3.562382 epochs/sec然后跑74 epochs，得到测试错误率为7.489%。