1.引言
一个多层感知机（Multi-Layer Perceptron，MLP）可以看做是，在逻辑回归分类器的中间加了非线性转换的隐层，这种转换把数据映射到一个线性可分的空间。一个单隐层的MLP就可以达到全局最优。
2.模型
一个单隐层的MLP可以表示如下：
一个隐层的MLP是一个函数：$f:R^{D}\rightarrow R^{L}$，其中 $D$ 是输入向量 $x$ 的大小，$L$是输出向量 $f(x)$ 的大小：
$f(x)=G(b^{(2)}+W^{(2)}(s(b^{(1)}+W^{(1)}))),$
向量$h(x)=s(b^{(1)}+W^{(1)})$构成了隐层，$W^{(1)}\in R^{D\times D_{h}}$ 是连接输入和隐层的权重矩阵，激活函数$s$可以是 $tanh(a)=(e^{a}-e^{-a})/(e^{a}+e^{-a})$ 或者 $sigmoid(a)=1/(1+e^{-a})$，但是前者通常会训练比较快。
在输出层得到：$o(x)=G(b^{(2)}+W^{(2)}h(x))$
为了训练MLP，所有参数 $\theta=\{W^{(2)},b^{(2)},W^{(1)},b^{\text{(1)}}\}.$ 用随机梯度下降法训练，参数的求导用反向传播算法来求。这里在顶层分类的时候用到了前面的逻辑回归的代码：
Python学习笔记之逻辑回归.
3.从逻辑回归到MLP
这里以单隐层MLP为例，当把数据由输入层映射到隐层之后，再加上一个逻辑回归层就构成了MLP.
class HiddenLayer(object):
def __init__(self, rng, input, n_in, n_out, W=None, b=None,
activation=T.tanh):
"""
Typical hidden layer of a MLP: units are fully-connected and have
sigmoidal activation function. Weight matrix W is of shape (n_in,n_out)
and the bias vector b is of shape (n_out,). NOTE : The nonlinearity used here is tanh Hidden unit activation is given by: tanh(dot(input,W) + b) :type rng: numpy.random.RandomState
:param rng: a random number generator used to initialize weights :type input: theano.tensor.dmatrix
:param input: a symbolic tensor of shape (n_examples, n_in) :type n_in: int
:param n_in: dimensionality of input :type n_out: int
:param n_out: number of hidden units :type activation: theano.Op or function
:param activation: Non linearity to be applied in the hidden
layer
"""
self.input = input
权重的初始化依赖于激活函数，根据[Xavier10]证明显示，对于$tanh$激活函数，权重初始值应该从$[-\sqrt{\frac{6}{fan_{in}+fan_{out}}},\sqrt{\frac{6}{fan_{in}+fan_{out}}}]$区间内均匀采样得到，其中 $fan_{in}$ 是第$(i-1)$ 层的单元数量，$fan_{out}$ 是第 $i$ 层的单元数量，对于sigmoid函数，采样区间应该变为 $[-4\sqrt{\frac{6}{fan_{in}+fan_{out}}},4\sqrt{\frac{6}{fan_{in}+fan_{out}}}]$.这种初始化方式能保证在训练的初始阶段，通过激活函数能够使得信息有效地向上和向下传播。
if W is None:
W_values = numpy.asarray(
rng.uniform(
# 随机数位于[low,high)区间
low=-numpy.sqrt(6. / (n_in + n_out)),
high=numpy.sqrt(6. / (n_in + n_out)),
size=(n_in, n_out)
),
# 类型设为 floatX 是为了在GPU上运行
dtype=theano.config.floatX
)
# 如果激活函数是 sigmoid，权重初始化要变大
if activation == theano.tensor.nnet.sigmoid:
W_values *= 4
# borrow = True 表示数据执行浅拷贝，增加效率
W = theano.shared(value=W_values, name='W', borrow=True) if b is None:
b_values = numpy.zeros((n_out,), dtype=theano.config.floatX)
b = theano.shared(value=b_values, name='b', borrow=True) self.W = W
self.b = b lin_output = T.dot(input, self.W) + self.b
self.output = (
lin_output if activation is None
else activation(lin_output)
)
# parameters of the model
self.params = [self.W, self.b]
在上面两步的基础上构建MLP:
class MLP(object):
"""Multi-Layer Perceptron Class A multilayer perceptron is a feedforward artificial neural network model
that has one layer or more of hidden units and nonlinear activations.
Intermediate layers usually have as activation function tanh or the
sigmoid function (defined here by a ``HiddenLayer`` class) while the
top layer is a softamx layer (defined here by a ``LogisticRegression``
class).
""" def __init__(self, rng, input, n_in, n_hidden, n_out):
"""Initialize the parameters for the multilayer perceptron :type rng: numpy.random.RandomState
:param rng: a random number generator used to initialize weights :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_hidden: int
:param n_hidden: number of hidden units :type n_out: int
:param n_out: number of output units, the dimension of the space in
which the labels lie """ # Since we are dealing with a one hidden layer MLP, this will translate
# into a HiddenLayer with a tanh activation function connected to the
# LogisticRegression layer; the activation function can be replaced by
# sigmoid or any other nonlinear function
self.hiddenLayer = HiddenLayer(
rng=rng,
input=input,
n_in=n_in,
n_out=n_hidden,
activation=T.tanh
) # The logistic regression layer gets as input the hidden units
# of the hidden layer
self.logRegressionLayer = LogisticRegression(
input=self.hiddenLayer.output,
n_in=n_hidden,
n_out=n_out
)
为了防止过拟合，这里加上 L1 和 L2 正则项，即计算权重 $W^{(1)},W^{(2)}$ 的1范数和2范数：
# L1 norm ; one regularization option is to enforce L1 norm to
# be small
self.L1 = (
abs(self.hiddenLayer.W).sum()
+ abs(self.logRegressionLayer.W).sum()
) # square of L2 norm ; one regularization option is to enforce
# square of L2 norm to be small
self.L2_sqr = (
(self.hiddenLayer.W ** 2).sum()
+ (self.logRegressionLayer.W ** 2).sum()
) # negative log likelihood of the MLP is given by the negative
# log likelihood of the output of the model, computed in the
# logistic regression layer
self.negative_log_likelihood = (
self.logRegressionLayer.negative_log_likelihood
)
# same holds for the function computing the number of errors
self.errors = self.logRegressionLayer.errors # the parameters of the model are the parameters of the two layer it is
# made out of
self.params = self.hiddenLayer.params + self.logRegressionLayer.params
似然函数的值加上正则项构成损失函数：
# the cost we minimize during training is the negative log likelihood of
# the model plus the regularization terms (L1 and L2); cost is expressed
# here symbolically
cost = (
classifier.negative_log_likelihood(y)
+ L1_reg * classifier.L1
+ L2_reg * classifier.L2_sqr
)
4.Minist识别测试
"""
This tutorial introduces the multilayer perceptron using Theano. A multilayer perceptron is a logistic regressor where
instead of feeding the input to the logistic regression you insert a
intermediate layer, called the hidden layer, that has a nonlinear
activation function (usually tanh or sigmoid) . One can use many such
hidden layers making the architecture deep. The tutorial will also tackle
the problem of MNIST digit classification. .. math:: f(x) = G( b^{(2)} + W^{(2)}( s( b^{(1)} + W^{(1)} x))), References: - textbooks: "Pattern Recognition and Machine Learning" -
Christopher M. Bishop, section 5 """
__docformat__ = 'restructedtext en' import os
import sys
import time import numpy import theano
import theano.tensor as T from logistic_sgd import LogisticRegression, load_data # start-snippet-1
class HiddenLayer(object):
def __init__(self, rng, input, n_in, n_out, W=None, b=None,
activation=T.tanh):
"""
Typical hidden layer of a MLP: units are fully-connected and have
sigmoidal activation function. Weight matrix W is of shape (n_in,n_out)
and the bias vector b is of shape (n_out,). NOTE : The nonlinearity used here is tanh Hidden unit activation is given by: tanh(dot(input,W) + b) :type rng: numpy.random.RandomState
:param rng: a random number generator used to initialize weights :type input: theano.tensor.dmatrix
:param input: a symbolic tensor of shape (n_examples, n_in) :type n_in: int
:param n_in: dimensionality of input :type n_out: int
:param n_out: number of hidden units :type activation: theano.Op or function
:param activation: Non linearity to be applied in the hidden
layer
"""
self.input = input
# end-snippet-1 # `W` is initialized with `W_values` which is uniformely sampled
# from sqrt(-6./(n_in+n_hidden)) and sqrt(6./(n_in+n_hidden))
# for tanh activation function
# the output of uniform if converted using asarray to dtype
# theano.config.floatX so that the code is runable on GPU
# Note : optimal initialization of weights is dependent on the
# activation function used (among other things).
# For example, results presented in [Xavier10] suggest that you
# should use 4 times larger initial weights for sigmoid
# compared to tanh
# We have no info for other function, so we use the same as
# tanh.
if W is None:
W_values = numpy.asarray(
rng.uniform(
# 随机数位于[low,high)区间
low=-numpy.sqrt(6. / (n_in + n_out)),
high=numpy.sqrt(6. / (n_in + n_out)),
size=(n_in, n_out)
),
# 类型设为 floatX 是为了在GPU上运行
dtype=theano.config.floatX
)
# 如果激活函数是 sigmoid，权重初始化要变大
if activation == theano.tensor.nnet.sigmoid:
W_values *= 4
# borrow = True 表示数据执行浅拷贝，增加效率
W = theano.shared(value=W_values, name='W', borrow=True) if b is None:
b_values = numpy.zeros((n_out,), dtype=theano.config.floatX)
b = theano.shared(value=b_values, name='b', borrow=True) self.W = W
self.b = b lin_output = T.dot(input, self.W) + self.b
self.output = (
lin_output if activation is None
else activation(lin_output)
)
# parameters of the model
self.params = [self.W, self.b] # start-snippet-2
class MLP(object):
"""Multi-Layer Perceptron Class A multilayer perceptron is a feedforward artificial neural network model
that has one layer or more of hidden units and nonlinear activations.
Intermediate layers usually have as activation function tanh or the
sigmoid function (defined here by a ``HiddenLayer`` class) while the
top layer is a softamx layer (defined here by a ``LogisticRegression``
class).
""" def __init__(self, rng, input, n_in, n_hidden, n_out):
"""Initialize the parameters for the multilayer perceptron :type rng: numpy.random.RandomState
:param rng: a random number generator used to initialize weights :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_hidden: int
:param n_hidden: number of hidden units :type n_out: int
:param n_out: number of output units, the dimension of the space in
which the labels lie """ # Since we are dealing with a one hidden layer MLP, this will translate
# into a HiddenLayer with a tanh activation function connected to the
# LogisticRegression layer; the activation function can be replaced by
# sigmoid or any other nonlinear function
self.hiddenLayer = HiddenLayer(
rng=rng,
input=input,
n_in=n_in,
n_out=n_hidden,
activation=T.tanh
) # The logistic regression layer gets as input the hidden units
# of the hidden layer
self.logRegressionLayer = LogisticRegression(
input=self.hiddenLayer.output,
n_in=n_hidden,
n_out=n_out
)
# end-snippet-2 start-snippet-3
# L1 norm ; one regularization option is to enforce L1 norm to
# be small
self.L1 = (
abs(self.hiddenLayer.W).sum()
+ abs(self.logRegressionLayer.W).sum()
) # square of L2 norm ; one regularization option is to enforce
# square of L2 norm to be small
self.L2_sqr = (
(self.hiddenLayer.W ** 2).sum()
+ (self.logRegressionLayer.W ** 2).sum()
) # negative log likelihood of the MLP is given by the negative
# log likelihood of the output of the model, computed in the
# logistic regression layer
self.negative_log_likelihood = (
self.logRegressionLayer.negative_log_likelihood
)
# same holds for the function computing the number of errors
self.errors = self.logRegressionLayer.errors # the parameters of the model are the parameters of the two layer it is
# made out of
self.params = self.hiddenLayer.params + self.logRegressionLayer.params
# end-snippet-3 def test_mlp(learning_rate=0.01, L1_reg=0.00, L2_reg=0.0001, n_epochs=1000,
dataset='mnist.pkl.gz', batch_size=20, n_hidden=500):
"""
Demonstrate stochastic gradient descent optimization for a multilayer
perceptron This is demonstrated on MNIST. :type learning_rate: float
:param learning_rate: learning rate used (factor for the stochastic
gradient :type L1_reg: float
:param L1_reg: L1-norm's weight when added to the cost (see
regularization) :type L2_reg: float
:param L2_reg: L2-norm's weight when added to the cost (see
regularization) :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
x = T.matrix('x') # the data is presented as rasterized images
y = T.ivector('y') # the labels are presented as 1D vector of
# [int] labels rng = numpy.random.RandomState(1234) # construct the MLP class
classifier = MLP(
rng=rng,
input=x,
n_in=28 * 28,
n_hidden=n_hidden,
n_out=10
) # start-snippet-4
# the cost we minimize during training is the negative log likelihood of
# the model plus the regularization terms (L1 and L2); cost is expressed
# here symbolically
cost = (
classifier.negative_log_likelihood(y)
+ L1_reg * classifier.L1
+ L2_reg * classifier.L2_sqr
)
# end-snippet-4 # 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]
}
) # start-snippet-5
# compute the gradient of cost with respect to theta (sotred in params)
# the resulting gradients will be stored in a list gparams
gparams = [T.grad(cost, param) for param in classifier.params] # specify how to update the parameters of the model as a list of
# (variable, update expression) pairs # given two list the zip A = [a1, a2, a3, a4] and B = [b1, b2, b3, b4] of
# same length, zip generates a list C of same size, where each element
# is a pair formed from the two lists :
# C = [(a1, b1), (a2, b2), (a3, b3), (a4, b4)]
updates = [
(param, param - learning_rate * gparam)
for param, gparam in zip(classifier.params, gparams)
] # 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-5 ###############
# TRAIN MODEL #
###############
print '... training' # early-stopping parameters
patience = 10000 # 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
best_iter = 0
test_score = 0.
start_time = time.clock() epoch = 0
done_looping = False
# 迭代 n_epochs 次，每次迭代都将遍历训练集所有样本
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 # 训练一定的样本之后才进行交叉验证
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
# 如果交叉验证的误差比当前最小的误差还小，就在测试集上测试
if this_validation_loss < best_validation_loss:
# improve patience if loss improvement is good enough
# 如果改善很多，就在本次迭代中多训练一定数量的样本
if (
this_validation_loss < best_validation_loss *
improvement_threshold
):
patience = max(patience, iter * patience_increase) # 记录最小的交叉验证误差和相应的迭代数
best_validation_loss = this_validation_loss
best_iter = iter # 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.))
# 训练样本数超过 patience，即停止
if patience <= iter:
done_looping = True
break end_time = time.clock()
print(('Optimization complete. Best validation score of %f %% '
'obtained at iteration %i, with test performance %f %%') %
(best_validation_loss * 100., best_iter + 1, test_score * 100.))
print >> sys.stderr, ('The code for file ' +
os.path.split(__file__)[1] +
' ran for %.2fm' % ((end_time - start_time) / 60.)) if __name__ == '__main__':
test_mlp()
关于上面代码中的交叉验证：只有训练结果的交叉验证结果比上一次交叉验证结果好，才在测试集上进行测试！
训练过程截图：
学习内容来源：