逻辑回归模型
逻辑回归是机器学习中做分类任务常用方法，属于“广义的线性模型”，即：
常用的输出函数（即g）包括sigmoid函数，可以参考：逻辑回归模型。上一章提到过，本例程的损失函数使用Negative log-likehood 对于NLL来说，常用输出函数为softmax函数。其定义如下：
可以看一下上式，对于某个样本(x,y)，该式代表了其属于第i个类别的概率。当其在第i个类别的响应越大于所有其他类别时，该值接近于1；当所有类别响应值接近时，其属于任何一个类别的概率都很小。
顺便提下，softmax函数也有很好的求导性质，softmax'(x) = softmax(x)(1-softmax(x))。下面证明一下，过程很简单：
因此，对于某一样本来说，其预测类别应该是使得softmax(wx+b)最大的i，即：
对应的代码如下，首先建立共享变量w和b，然后根据上述公式计算Ypred:
# 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)
代价函数
上一小节，是在得到模型参数w,b的基础上预测数据，那么模型参数是如何得来的呢，即训练模型过程。在热身训练中已经讲述过如何训练模型，即通过写出代价函数以及梯度公式，使用批量梯度下降法，这里做简要回顾。
对应代码如下，注释很多，就最后一行是有用的。这行代码的意思就是求v的均值，v= [ LP[0,y[0]]，LP[1,y[1]]，...L[n-1,y[n-1]] ]。LP[0,y[0]]的含义就是第0个样本属于其真实类别y[0]的概率（0~1）。原公式是求和，这里是求和的均值，为了是不受不同算法中批量数据数目（即n）不同的影响。
# 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])
创建逻辑回归类
下面就可以根据之前描述的定义一个逻辑回归类了，包括模型参数（w,b），输入个数和输出个数(n_in,n_out)；此外还添加了计算错误率函数errors，找出预测结果和实际结果不等数目，同样取均值还是为了不受个数n影响（即求错误率）。
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
"""
# 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
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)
# 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 -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', 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()
初始化该类的方法，以mnist数据类型为例（输入28*28，输出0~9）。
# 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)
需要优化的损失函数变量cost可通过类来获得：
# the cost we minimize during training is the negative log likelihood of
# the model in symbolic format
cost = classifier.negative_log_likelihood(y)
训练模型
除了提供损失函数外，还需要提供变量导数，Theano提供了简单获得导数方法：
g_W = T.grad(cost=cost, wrt=classifier.W) g_b = T.grad(cost=cost, wrt=classifier.b)
同时定义相应的训练函数：
# 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]
}
)
每一次train_model函数被调用，其都会计算当前batch的损失函数，并且进行一次梯度下降，通过输入index来调整不同batch数据。
测试模型
在热身训练中，提到过为了防止模型过拟合，通常采用两种方法，一是加入正则化项，二是测试模型对交叉验证集的性能，如果模型在优化过程中不能在交叉验证集上获得更优的性能，则停止优化。定义交叉验证优化函数和训练优化函数一样，只是执行的频率不同：
# 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]
}
)
而模型的性能就是通过错误率（函数errors）来评价，这在上面已经提到过了。
总结
整合以上所有知识，就可以用逻辑模型对mnist数据集操作了，回顾一下核心内容：
（1）使用softmax作为输出函数（即预测函数），NLL模型作为代价函数，错误率作为实际模型性能评价（注意NLL下降可不代表错误率下降）
（2）优化采用批量梯度下降法（样例中batch_size = 600）
（3）为防止过拟合，每隔一定优化次数，验证交叉训练集的错误率也在下降
在我的GTX960显卡下，最终运行结果如下：
运行代码，看着很长，其实很多注释，核心内容都在本篇文章内：
"""
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
"""
from __future__ import print_function
__docformat__ = 'restructedtext en'
import six.moves.cPickle as pickle
import gzip
import os
import sys
import timeit
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 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} (\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':
from six.moves import urllib
origin = (
'http://www.iro.umontreal.ca/~lisa/deep/data/mnist/mnist.pkl.gz'
)
print('Downloading data from %s' % origin)
urllib.request.urlretrieve(origin, dataset)
print('... loading data')
# Load the dataset
with gzip.open(dataset, 'rb') as f:
try:
train_set, valid_set, test_set = pickle.load(f, encoding='latin1')
except:
train_set, valid_set, test_set = pickle.load(f)
# train_set, valid_set, test_set format: tuple(input, target)
# input is a numpy.ndarray of 2 dimensions (a matrix)
# where each row corresponds to an example. target is a
# numpy.ndarray of 1 dimension (vector) that has the same length as
# the number of rows in the input. It should give the 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 = timeit.default_timer()
done_looping = False
epoch = 0
while (epoch < n_epochs) and (not done_looping):
epoch = epoch + 1
for minibatch_index in range(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 range(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 range(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.
)
)
# save the best model
with open('best_model.pkl', 'wb') as f:
pickle.dump(classifier, f)
if patience <= iter:
done_looping = True
break
end_time = timeit.default_timer()
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(('The code for file ' +
os.path.split(__file__)[1] +
' ran for %.1fs' % ((end_time - start_time))), file=sys.stderr)
def predict():
"""
An example of how to load a trained model and use it
to predict labels.
"""
# load the saved model
classifier = pickle.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)
if __name__ == '__main__':
sgd_optimization_mnist()
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