简介
在这节我们使用Theano用于最基本的分类器：Logistic回归(Logistic Regression)。
全部的代码可以在我的CSDN下载中免费下载：http://download.csdn.net/detail/ws_20100/9222263。
下面我们从模型开始。
模型
逻辑回归是一个概率，线性分类器。它的参数包含一个权值矩阵
在数学上，一个输入向量
# 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 biases 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 hyperplane 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
# hyperplane-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)由于模型的参数在训练时始终要保持回归状态，因此我们将p_y_given_x是一个向量类型的符号变量。为了得到实际模型的预测值，我们使用T.argmax运算符，这将返回一个索引值，代表在p_y_given_x的哪个位置的值最大(例如，具有最大概率的类别)。
现在，我们所定义的模型其实并不能做任何事情，因为所有的参数都处于初始状态，下面将会介绍，如何学习最优化参数。
定义损失函数
训练最优化模型参数，涉及到最小化损失函数。在多类别的分类问题中，很自然的想法是使用负对数似然作为损失函数。这等价于在参数为
这里使用的是小批量的概率梯度方法(mini-batches Stochastic Gradient Descent, MSGD)。Theano代码中关于损失的定义如下：
# 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])尽管在格式上，损失函数定义为数据集上每个误差项和的形式。但是在代码中，使用的是平均函数
(T.mean)，因为这样可以使得学习率不依赖于数据批的大小。
创建一个回归类
我们现在要定义一个LogisticRegression类，来囊括逻辑回归的所有特征。下面的代码包含了许多方面，并且注释的很清楚哦。
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 biases 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 hyperplane 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
# hyperplane-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]
# keep track of model input
self.input = input
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} ( heta={W,b}, mathcal{D}) =
frac{1}{|mathcal{D}|} sum_{i=0}^{|mathcal{D}|}
log(P(Y=y^{(i)}|x^{(i)}, W,b)) \
ell ( heta={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()那么如何来实例化一个LogisticRegression类呢，可以看如下代码：
# 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)在上面的代码中，首先定义了输入变量x和y是定义在LogisticRegression对象以外的。因为这个类需要输入值作为它__init__函数的参数，当你希望连接这些实例，来组成深度网络时，这个设定非常有用。一个层的输出可以当作下一层的输入。（这里并没有构建多层网络，但是代码可以在多层网络中重用）
最后，我们定义一个（符号化）cost变量，用来最小化，使用实例方法classifier.negative_log_likelihood。
# the cost we minimize during training is the negative log likelihood of
# the model in symbolic format
cost = classifier.negative_log_likelihood(y)注意定义cost中有一个隐含的符号输入x，因为classifier的符号变量在初始化时就定义在x中。
学习模型
在多数编程语言（C/C++，Matlab，Python）中实现MSGD，可以使用损失函数对于参数的梯度表达式：
使用Theano，这种工作被大量简化。它可以完成自动求导，并应用相应的数学变换，以提高数值稳定性。
在Theano中获得
g_W = T.grad(cost=cost, wrt=classifier.W)
g_b = T.grad(cost=cost, wrt=classifier.b)g_W和g_b是符号变量，可以用于计算。函数train_model可以完成梯度下降，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]
}
)updates是一系列二元组。在每个二元组中，第一个元素是待更新的符号变量，第二个元素是用于计算新数值的符号函数。相似地，givens是一个字典，其中关键字是符号变量，其中的值指定了它们的置换。函数train_model定义如下：
- 输入是小批量数据的索引
index，连带数据批的大小(这不是输入，因为它是一个固定值)，定义了x 和相应的标签y 。 - 返回值是
index对应的x 和y 所计算出的代价/损失值。 - 对于每个函数调用，首先根据
index置换训练集中x和y的值。然后，将会评估该数据批所对应的代价值，并应用updates更新。
每次调用train_model(index)，它将会计算数据批，返回其代价值，并完成了MSGD的一步。整个的学习算法会不断地循环遍历所有的数据集，并且一次只会考虑一批数据内的所有样本，然后重复地调用train_model函数。
测试模型
当测试模型的时候，我们注重于错误分类的样本个数。所以，LogisticRegression类有一个额外的实例方法，用于尝试减少每个数据批中错误分类的样本个数。代码如下：
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()然后，我们创建了一个函数test_model和一个函数validate_model，我们可以调用这些函数来挽回错误分类的值。你将会看到，validate_model是迭代退出的关键。这些函数的输入参数是数据批的索引号，然后函数将计算该索引号所对应的数据批中错误分类的个数。两个函数的唯一区别在于test_model面向的是测试集，validate_model面向的是验证集。
# 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]
}
)综合代码
最后的代码如下：【下载地址：http://download.csdn.net/detail/ws_20100/9222263】
使用者可以通过输入以下命令，使用SGD逻辑回归对MNIST字符进行分类。
python code/logistic_sgd.py输出应该是这样的形式：
...
epoch 72, minibatch 83/83, validation error 7.510417 %
epoch 72, minibatch 83/83, test error of best model 7.510417 %
epoch 73, minibatch 83/83, validation error 7.500000 %
epoch 73, minibatch 83/83, test error of best model 7.489583 %
Optimization complete with best validation score of 7.500000 %,with test performance 7.489583 %
The code run for 74 epochs, with 1.936983 epochs/sec在一个Intel酷睿双核CPU E8400 @3.00 Ghz的主机上，大约1.936 epochs/sec，在经历75 epochs后，测试误差为7.489%。在GPU上，大约10.0 epochs/sec。
使用已训练的模型进行预测
当训练达到最低误差的时候，我们可以重新载入模型对新数据的标签进行预测，predict函数完成了这些操作：
def predict():
"""
An example of how to load a trained model and use it
to predict labels.
"""
# load the saved model
classifier = cPickle.load(open('best_model.pkl'))
# compile a predictor function
predict_model = theano.function(
inputs=[classifier.input],
outputs=classifier.y_pred)
# We can test it on some examples from test test
dataset='mnist.pkl.gz'
datasets = load_data(dataset)
test_set_x, test_set_y = datasets[2]
test_set_x = test_set_x.get_value()
predicted_values = predict_model(test_set_x[:10])
print ("Predicted values for the first 10 examples in test set:")
print predicted_values参考资料
Theano深度学习资料：http://deeplearning.net/tutorial/logreg.html
【脚注】
[1]对于更小的数据集或更简单的模型，复杂的下降算法可能更加有效。logistic_cg.py代码阐述了如何使用SciPy的共扼梯度方法(conjugate gradient solver)完成逻辑回归任务。logistic_cg.py代码可以在我的CSDN下载中免费下载：http://download.csdn.net/detail/ws_20100/9223959