logistic regression using Theano and stochastic gradient descent
本文是学习Theno教程做的网页版笔记！后续的学习过程中会重点关注如下的网站：
- http://deeplearning.net/tutorial/logreg.html
- https://www.microsoft.com/en-us/research/uploads/prod/2006/01/Bishop-Pattern-Recognition-and-Machine-Learning-2006.pdf
1.工程目录组织如下:
数据下载地址：
http://www.iro.umontreal.ca/~lisa/deep/data/mnist/mnist.pkl.gz
2.程序代码
# -*- 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
"""
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='C:/Users/Administrator/ATC20181123/LogisticRegression1/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('C:/Users/Administrator/ATC20181123/LogisticRegression1/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('C:/Users/Administrator/ATC20181123/LogisticRegression1/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='C:/Users/Administrator/ATC20181123/LogisticRegression1/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()
“”“#测试取定不同的batch_size时sgd的变化情况
if __name__ == '__main__':
batch_size=[100,200,500,1000,2000,3000,5000]
for i in batch_size:
print("=======================batch_size======================")
print('batch_size= %d : ' % i)
print("=======================batch_size======================")
sgd_optimization_mnist(batch_size=i)
”“”
loading data
building the model
training the model
epoch 1, minibatch 83/83, validation error 12.458333 %
epoch 1, minibatch 83/83, test error of best model 12.375000 %
epoch 2, minibatch 83/83, validation error 11.010417 %
epoch 2, minibatch 83/83, test error of best model 10.958333 %
epoch 3, minibatch 83/83, validation error 10.312500 %
epoch 3, minibatch 83/83, test error of best model 10.312500 %
epoch 4, minibatch 83/83, validation error 9.875000 %
epoch 4, minibatch 83/83, test error of best model 9.833333 %
epoch 5, minibatch 83/83, validation error 9.562500 %
epoch 5, minibatch 83/83, test error of best model 9.479167 %
epoch 6, minibatch 83/83, validation error 9.322917 %
epoch 6, minibatch 83/83, test error of best model 9.291667 %
epoch 7, minibatch 83/83, validation error 9.187500 %
epoch 7, minibatch 83/83, test error of best model 9.000000 %
epoch 8, minibatch 83/83, validation error 8.989583 %
epoch 8, minibatch 83/83, test error of best model 8.958333 %
epoch 9, minibatch 83/83, validation error 8.937500 %
epoch 9, minibatch 83/83, test error of best model 8.812500 %
epoch 10, minibatch 83/83, validation error 8.750000 %
epoch 10, minibatch 83/83, test error of best model 8.666667 %
epoch 11, minibatch 83/83, validation error 8.666667 %
epoch 11, minibatch 83/83, test error of best model 8.520833 %
epoch 12, minibatch 83/83, validation error 8.583333 %
epoch 12, minibatch 83/83, test error of best model 8.416667 %
epoch 13, minibatch 83/83, validation error 8.489583 %
epoch 13, minibatch 83/83, test error of best model 8.291667 %
epoch 14, minibatch 83/83, validation error 8.427083 %
epoch 14, minibatch 83/83, test error of best model 8.281250 %
epoch 15, minibatch 83/83, validation error 8.354167 %
epoch 15, minibatch 83/83, test error of best model 8.270833 %
epoch 16, minibatch 83/83, validation error 8.302083 %
epoch 16, minibatch 83/83, test error of best model 8.239583 %
epoch 17, minibatch 83/83, validation error 8.250000 %
epoch 17, minibatch 83/83, test error of best model 8.177083 %
epoch 18, minibatch 83/83, validation error 8.229167 %
epoch 18, minibatch 83/83, test error of best model 8.062500 %
epoch 19, minibatch 83/83, validation error 8.260417 %
epoch 20, minibatch 83/83, validation error 8.260417 %
epoch 21, minibatch 83/83, validation error 8.208333 %
epoch 21, minibatch 83/83, test error of best model 7.947917 %
epoch 22, minibatch 83/83, validation error 8.187500 %
epoch 22, minibatch 83/83, test error of best model 7.927083 %
epoch 23, minibatch 83/83, validation error 8.156250 %
epoch 23, minibatch 83/83, test error of best model 7.958333 %
epoch 24, minibatch 83/83, validation error 8.114583 %
epoch 24, minibatch 83/83, test error of best model 7.947917 %
epoch 25, minibatch 83/83, validation error 8.093750 %
epoch 25, minibatch 83/83, test error of best model 7.947917 %
epoch 26, minibatch 83/83, validation error 8.104167 %
epoch 27, minibatch 83/83, validation error 8.104167 %
epoch 28, minibatch 83/83, validation error 8.052083 %
epoch 28, minibatch 83/83, test error of best model 7.843750 %
epoch 29, minibatch 83/83, validation error 8.052083 %
epoch 30, minibatch 83/83, validation error 8.031250 %
epoch 30, minibatch 83/83, test error of best model 7.843750 %
epoch 31, minibatch 83/83, validation error 8.010417 %
epoch 31, minibatch 83/83, test error of best model 7.833333 %
epoch 32, minibatch 83/83, validation error 7.979167 %
epoch 32, minibatch 83/83, test error of best model 7.812500 %
epoch 33, minibatch 83/83, validation error 7.947917 %
epoch 33, minibatch 83/83, test error of best model 7.739583 %
epoch 34, minibatch 83/83, validation error 7.875000 %
epoch 34, minibatch 83/83, test error of best model 7.729167 %
epoch 35, minibatch 83/83, validation error 7.885417 %
epoch 36, minibatch 83/83, validation error 7.843750 %
epoch 36, minibatch 83/83, test error of best model 7.697917 %
epoch 37, minibatch 83/83, validation error 7.802083 %
epoch 37, minibatch 83/83, test error of best model 7.635417 %
epoch 38, minibatch 83/83, validation error 7.812500 %
epoch 39, minibatch 83/83, validation error 7.812500 %
epoch 40, minibatch 83/83, validation error 7.822917 %
epoch 41, minibatch 83/83, validation error 7.791667 %
epoch 41, minibatch 83/83, test error of best model 7.625000 %
epoch 42, minibatch 83/83, validation error 7.770833 %
epoch 42, minibatch 83/83, test error of best model 7.614583 %
epoch 43, minibatch 83/83, validation error 7.750000 %
epoch 43, minibatch 83/83, test error of best model 7.593750 %
epoch 44, minibatch 83/83, validation error 7.739583 %
epoch 44, minibatch 83/83, test error of best model 7.593750 %
epoch 45, minibatch 83/83, validation error 7.739583 %
epoch 46, minibatch 83/83, validation error 7.739583 %
epoch 47, minibatch 83/83, validation error 7.739583 %
epoch 48, minibatch 83/83, validation error 7.708333 %
epoch 48, minibatch 83/83, test error of best model 7.583333 %
epoch 49, minibatch 83/83, validation error 7.677083 %
epoch 49, minibatch 83/83, test error of best model 7.572917 %
epoch 50, minibatch 83/83, validation error 7.677083 %
epoch 51, minibatch 83/83, validation error 7.677083 %
epoch 52, minibatch 83/83, validation error 7.656250 %
epoch 52, minibatch 83/83, test error of best model 7.541667 %
epoch 53, minibatch 83/83, validation error 7.656250 %
epoch 54, minibatch 83/83, validation error 7.635417 %
epoch 54, minibatch 83/83, test error of best model 7.520833 %
epoch 55, minibatch 83/83, validation error 7.635417 %
epoch 56, minibatch 83/83, validation error 7.635417 %
epoch 57, minibatch 83/83, validation error 7.604167 %
epoch 57, minibatch 83/83, test error of best model 7.489583 %
epoch 58, minibatch 83/83, validation error 7.583333 %
epoch 58, minibatch 83/83, test error of best model 7.458333 %
epoch 59, minibatch 83/83, validation error 7.572917 %
epoch 59, minibatch 83/83, test error of best model 7.468750 %
epoch 60, minibatch 83/83, validation error 7.572917 %
epoch 61, minibatch 83/83, validation error 7.583333 %
epoch 62, minibatch 83/83, validation error 7.572917 %
epoch 62, minibatch 83/83, test error of best model 7.520833 %
epoch 63, minibatch 83/83, validation error 7.562500 %
epoch 63, minibatch 83/83, test error of best model 7.510417 %
epoch 64, minibatch 83/83, validation error 7.572917 %
epoch 65, minibatch 83/83, validation error 7.562500 %
epoch 66, minibatch 83/83, validation error 7.552083 %
epoch 66, minibatch 83/83, test error of best model 7.520833 %
epoch 67, minibatch 83/83, validation error 7.552083 %
epoch 68, minibatch 83/83, validation error 7.531250 %
epoch 68, minibatch 83/83, test error of best model 7.520833 %
epoch 69, minibatch 83/83, validation error 7.531250 %
epoch 70, minibatch 83/83, validation error 7.510417 %
epoch 70, minibatch 83/83, test error of best model 7.500000 %
epoch 71, minibatch 83/83, validation error 7.520833 %
epoch 72, minibatch 83/83, validation error 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 5.068653 epochs/sec
The code for file wu2.py ran for 14.6s
3.如何将控制台打印的信息保存到文档，方便查看和比较？
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
#记录下来控制台的输出内容到日志中
import sys
class Logger(object):
def __init__(self, fileN="Default.log"):
self.terminal = sys.stdout
self.log = open(fileN, "w+")
def write(self, message):
self.terminal.write(message)
self.log.write(message)
def flush(self):
pass
sys.stdout = Logger("C:/Users/Administrator/Desktop//sgd.txt") # 保存到桌面
class LogisticRegression(object):
……
#if __name__ == '__main__':
# sgd_optimization_mnist()
if __name__ == '__main__':
batch_size=[100,200,500,1000,2000,3000,5000]
for i in batch_size:
print("=======================batch_size======================")
print('batch_size= %d : ' % i)
print("=======================batch_size======================")
sgd_optimization_mnist(batch_size=i)