基于theano的深度卷积神经网络

1.引言

卷积神经网络（Convolutional Neural Networks , CNN）受到视网膜上的细胞只对视野范围内的部分区域敏感，这一部分区域称为感受域（receptive field）.卷积神经网络正是采用了这种机制，每一个神经元只与一部分输入相连接。

2.稀疏连接

CNNs通过局部连接的方式揭示了空间中的局部相关性。在 $m$ 层的隐单元的输入来自于 $m-1$ 层的一部分单元的加权和，这一部分单元在空间上是连续的感受域。如下图：

可以把 $m-1$ 层想象成视网膜输入。$m$ 层的单元的感受域的宽度均为3，因此只与视网膜层的 3 个相邻的神经元相连接。$m+1$ 层的单元与其下面一层的连接方式也是如此。每一个神经元对不在感受域范围内的变化是没有反应的，所以上面的结构保证学习出一种“滤波器“，使其对局部空间的输入模式产生强烈的反应。

但是，正如上面图中所示，把许多这样的滤波器层层级联，局部感知逐渐变得全局感知，$m$ 层的每一个单元只对部分输入感知，而 $m+1$ 层的单元又将 $m$ 层的感知结果综合起来从而形成对输入层全部的一个感知，所以$m+1$隐层单元可以看作是对宽度为5的特征的一个非线性编码。

3.共享权重（Shared Weights）

在CNNs，每个滤波器 $h_{i}$ 重复地逐步横跨整个输入层。重复的单元共享参数（权重向量和偏置），从而形成一幅特征图。

在上图中，3个隐层单元属于同一幅特征图，一样颜色的权重值是共享的，即相等的。

滤波器通过这种方式使得图像中可视层中任意位置的特征都能被检测出来，权重共享大大减少了需要学习的参数的数量。

4.细节和符号

通过重复地把一个函数运用到整个图像的子区域可以得到一幅特征图，即用一个线性滤波器对图像进行卷积操作,加上偏置项，然后再采用一个非线性函数。如果用 $h^{k}$ 表示第 $k$ 幅特征图，其对应的滤波器由 $W^{k}$ 和偏置 $b_{k}$ 决定, 那么特征图 $h^{k}$ 可以由下计算得到（采用 tanh 作为非线性函数）：

$h_{ij}^{k}=tanh((W^{k}*x)_{ij}+b_{k}$

为了得到对数据更加丰富的表示，通常每个隐层都由多幅特征图组成:${h^{ ext{(k)}},k=0,...K}$.权重 $W$ 由一个4维的张量表示, 4各维度分别表示：目的特征图，源特征图，源特征图的垂直坐标，源特征图的水平坐标。偏置 $b$ 由一个向量表示，其中每一个元素是每一个目标特征图对应的偏置。可以表示如下：

在上图中 $W_{ij}^{kl}$ 表示在 $m-1$ 层的第 $k$ 幅特征图的每一个像素 与第 $m$ 层的第 $l$ 幅特征图的像素 $(i,j)$ 之间的连接权重。

5.卷积操作

卷积操作（Convolution operation,ConvOp）在theano中是通过theano.tensor.signal.conv2d实现的，它需要两个输入：

• 输入图像的部分子集对应的一个4阶张量，该张量的每一维分别表示：子集的大小，输入特征图的编号，图像的高度，图像的宽度
• 表示权重矩阵 $W$ 的一个4阶张量，每一维分别表示：在 $m$ 的特征图像的编号，$m-1$ 层特征图像的编号，滤波器的高度，滤波器的宽度

这里还要介绍一个在下面代码中将要用到的一个函数 dimshuffle(*pattern):

　　例如dimshuffle('x', 2, 'x', 0, 1)，就是将原来3阶张量扩展为5阶张量，新张量的第0维和第2维为0，而第1维，第3维和第4维分别由原来3阶张量的第2维，第0维和第1维映射而来。

　　如果原来张量的形状为（20,30,40），通过dimshuffle('x', 2, 'x', 0, 1)之后，形状变为（1,40,1,20,30）

　　dimshuffle(0, 1) -> 和原来一样

　　dimshuffle(1, 0) -> 交换第1维和第0维的数据

　　更多详细资料参看：dimshuffle

下面用到的图片3wolfmoon

下面对输入是3 幅RGB 特征图，进行卷积操作，并输出卷积前后的对比图：

# -*- coding: utf-8 -*-
"""
Created on Tue Apr 28 10:22:14 2015

@author: ZengJiulin
"""

import theano
from theano import tensor as T
from theano.tensor.nnet import conv
import pylab
from PIL import Image
import numpy

rng = numpy.random.RandomState(23455)

# instantiate 4D tensor for input
input = T.tensor4(name='input',dtype='float64')

# initialize shared variable for weights.
# 输出的特征图 2 幅
# 输入的特征图 3 幅
# 滤波器的大小 9*9
w_shp = (2, 3, 9, 9)
w_bound = numpy.sqrt(3 * 9 * 9)
W = theano.shared( numpy.asarray(
            rng.uniform(
                low=-1.0 / w_bound,
                high=1.0 / w_bound,
                size=w_shp),
            dtype=input.dtype), name ='W')

# initialize shared variable for bias (1D tensor) with random values
# IMPORTANT: biases are usually initialized to zero. However in this
# particular application, we simply apply the convolutional layer to
# an image without learning the parameters. We therefore initialize
# them to random values to "simulate" learning.
# 输出的特征图有 2 幅，所以偏置向量的元素个数同样为 2
b_shp = (2,)
b = theano.shared(numpy.asarray(
            rng.uniform(low=-.5, high=.5, size=b_shp),
            dtype=input.dtype), name ='b')

# build symbolic expression that computes the convolution of input with filters in w
conv_out = conv.conv2d(input, W)

# build symbolic expression to add bias and apply activation function, i.e. produce neural net layer output
# A few words on ``dimshuffle`` :
#   ``dimshuffle`` is a powerful tool in reshaping a tensor;
#   what it allows you to do is to shuffle dimension around
#   but also to insert new ones along which the tensor will be
#   broadcastable;
#   dimshuffle('x', 2, 'x', 0, 1)
#   This will work on 3d tensors with no broadcastable
#   dimensions. The first dimension will be broadcastable,
#   then we will have the third dimension of the input tensor as
#   the second of the resulting tensor, etc. If the tensor has
#   shape (20, 30, 40), the resulting tensor will have dimensions
#   (1, 40, 1, 20, 30). (AxBxC tensor is mapped to 1xCx1xAxB tensor)
#   More examples:
#    dimshuffle('x') -> make a 0d (scalar) into a 1d vector
#    dimshuffle(0, 1) -> identity
#    dimshuffle(1, 0) -> inverts the first and second dimensions
#    dimshuffle('x', 0) -> make a row out of a 1d vector (N to 1xN)
#    dimshuffle(0, 'x') -> make a column out of a 1d vector (N to Nx1)
#    dimshuffle(2, 0, 1) -> AxBxC to CxAxB
#    dimshuffle(0, 'x', 1) -> AxB to Ax1xB
#    dimshuffle(1, 'x', 0) -> AxB to Bx1xA

# 卷积后的结果加上偏置，然后进行一个非线性函数计算，这里采用的是sigmoid函数
output = T.nnet.sigmoid(conv_out + b.dimshuffle('x', 0, 'x', 'x'))

# create theano function to compute filtered images
f = theano.function([input], output)



# open random image of dimensions 639x516
img_file = open('E:\Python\3wolfmoon.jpg','rb')
img = Image.open(img_file)
# dimensions are (height, width, channel)
img = numpy.asarray(img, dtype='float64') / 256.

# put image in 4D tensor of shape (1, 3, height, width)
cc = img.transpose(2, 0, 1)
img_ = img.transpose(2, 0, 1).reshape(1, 3, 639, 516)
filtered_img = f(img_)

# plot original image and first and second components of output
pylab.subplot(1, 3, 1); pylab.axis('off'); pylab.imshow(img)
pylab.gray();
# recall that the convOp output (filtered image) is actually a "minibatch",
# of size 1 here, so we take index 0 in the first dimension:
pylab.subplot(1, 3, 2); pylab.axis('off'); pylab.imshow(filtered_img[0, 0, :, :])
pylab.subplot(1, 3, 3); pylab.axis('off'); pylab.imshow(filtered_img[0, 1, :, :])
pylab.show()

注意到，随机初始化的滤波器非常像一个边缘检测器。

6.最大池化（MaxPooling）

最大池化是一种下采样的形式，最大池化额操作就是把图像分割成不重叠的矩形区域，每一个子区域选出一个最大值。

最大池化的两个作用：

• 去除了非最大值，减少了后面一层的计算量
• (这里还没怎么看懂，后面是原讲义的说法)It provides a form of translation invariance. Imagine cascading a max-pooling layer with a convolutional layer. There are 8 directions in which one can translate the input image by a single pixel. If max-pooling is done over a 2x2 region, 3 out of these 8 possible configurations will produce exactly the same output at the convolutional layer. For max-pooling over a 3x3 window, this jumps to 5/8.Since it provides additional robustness to position, max-pooling is a “smart” way of reducing the dimensionality of intermediate representations.

最大池化在theano中是通过theano.tensor.signal.downsample.max_pool_2d实现的，例如：

# -*- coding: utf-8 -*-
"""
Created on Tue Apr 28 15:17:23 2015

@author: ZengJiulin
"""
import theano
from theano import tensor as T
import numpy
from theano.tensor.signal import downsample

input = T.dtensor4('input')
maxpool_shape = (2, 2)
pool_out = downsample.max_pool_2d(input, maxpool_shape, ignore_border=True)
f = theano.function([input],pool_out)

invals = numpy.random.RandomState(1).rand(3, 2, 5, 5)
print 'With ignore_border set to True:'
print 'invals[0, 0, :, :] =
', invals[0, 0, :, :]
print 'output[0, 0, :, :] =
', f(invals)[0, 0, :, :]

pool_out = downsample.max_pool_2d(input, maxpool_shape, ignore_border=False)
f = theano.function([input],pool_out)
print 'With ignore_border set to False:'
print 'invals[1, 0, :, :] =
 ', invals[1, 0, :, :]
print 'output[1, 0, :, :] =
 ', f(invals)[1, 0, :, :]

注意忽略边界和不忽略边界的区别：

>>> runfile('E:/Python/downsample.py', wdir=r'E:/Python')
Using gpu device 0: GeForce GT 720M
With ignore_border set to True:
invals[0, 0, :, :] =
[[  4.17022005e-01   7.20324493e-01   1.14374817e-04   3.02332573e-01
    1.46755891e-01]
 [  9.23385948e-02   1.86260211e-01   3.45560727e-01   3.96767474e-01
    5.38816734e-01]
 [  4.19194514e-01   6.85219500e-01   2.04452250e-01   8.78117436e-01
    2.73875932e-02]
 [  6.70467510e-01   4.17304802e-01   5.58689828e-01   1.40386939e-01
    1.98101489e-01]
 [  8.00744569e-01   9.68261576e-01   3.13424178e-01   6.92322616e-01
    8.76389152e-01]]
output[0, 0, :, :] =
[[ 0.72032449  0.39676747]
 [ 0.6852195   0.87811744]]
With ignore_border set to False:
invals[1, 0, :, :] =
  [[ 0.01936696  0.67883553  0.21162812  0.26554666  0.49157316]
 [ 0.05336255  0.57411761  0.14672857  0.58930554  0.69975836]
 [ 0.10233443  0.41405599  0.69440016  0.41417927  0.04995346]
 [ 0.53589641  0.66379465  0.51488911  0.94459476  0.58655504]
 [ 0.90340192  0.1374747   0.13927635  0.80739129  0.39767684]]
output[1, 0, :, :] =
  [[ 0.67883553  0.58930554  0.69975836]
 [ 0.66379465  0.94459476  0.58655504]
 [ 0.90340192  0.80739129  0.39767684]]
>>> 

7.LeNet整个模型

稀疏，卷积层和最大池化是 LeNet 模型的核心，但是具体的其他细节可能变化很大。下图给出LeNet的一个描述：

底层由卷积层和下采样层交替，顶层与传统的 MLP 全连接。

从整个执行过程看，就是把一个4阶的张量整理成MLP能够处理的2维特征图。

8.全部代码

# -*- coding: utf-8 -*-
"""
Created on Sat Apr 25 14:20:02 2015

@author: ZengJiulin
"""

"""This tutorial introduces the LeNet5 neural network architecture
using Theano.  LeNet5 is a convolutional neural network, good for
classifying images. This tutorial shows how to build the architecture,
and comes with all the hyper-parameters you need to reproduce the
paper's MNIST results.


This implementation simplifies the model in the following ways:

 - LeNetConvPool doesn't implement location-specific gain and bias parameters
 - LeNetConvPool doesn't implement pooling by average, it implements pooling
   by max.
 - Digit classification is implemented with a logistic regression rather than
   an RBF network
 - LeNet5 was not fully-connected convolutions at second layer

References:
 - Y. LeCun, L. Bottou, Y. Bengio and P. Haffner:
   Gradient-Based Learning Applied to Document
   Recognition, Proceedings of the IEEE, 86(11):2278-2324, November 1998.
   http://yann.lecun.com/exdb/publis/pdf/lecun-98.pdf

"""
import os
import sys
import time

import numpy

import theano
import theano.tensor as T
from theano.tensor.signal import downsample
from theano.tensor.nnet import conv

from logistic_sgd import LogisticRegression, load_data
from mlp import HiddenLayer


class LeNetConvPoolLayer(object):
    """Pool Layer of a convolutional network """

    def __init__(self, rng, input, filter_shape, image_shape, poolsize=(2, 2)):
        """
        Allocate a LeNetConvPoolLayer with shared variable internal parameters.

        :type rng: numpy.random.RandomState
        :param rng: a random number generator used to initialize weights

        :type input: theano.tensor.dtensor4
        :param input: symbolic image tensor, of shape image_shape

        :type filter_shape: tuple or list of length 4
        :param filter_shape: (number of filters, num input feature maps,
                              filter height, filter width)

        :type image_shape: tuple or list of length 4
        :param image_shape: (batch size, num input feature maps,
                             image height, image width)

        :type poolsize: tuple or list of length 2
        :param poolsize: the downsampling (pooling) factor (#rows, #cols)
        """

        assert image_shape[1] == filter_shape[1]
        self.input = input

        # there are "num input feature maps * filter height * filter width"
        # inputs to each hidden unit
        
        fan_in = numpy.prod(filter_shape[1:])
        # each unit in the lower layer receives a gradient from:
        # "num output feature maps * filter height * filter width" /
        #   pooling size
        fan_out = (filter_shape[0] * numpy.prod(filter_shape[2:]) /
                   numpy.prod(poolsize))
        # initialize weights with random weights
        W_bound = numpy.sqrt(6. / (fan_in + fan_out))
        #卷积核本质上就是下面这个权重矩阵
        self.W = theano.shared(
            numpy.asarray(
                rng.uniform(low=-W_bound, high=W_bound, size=filter_shape),
                dtype=theano.config.floatX
            ),
            borrow=True
        )

        # the bias is a 1D tensor -- one bias per output feature map
        b_values = numpy.zeros((filter_shape[0],), dtype=theano.config.floatX)
        self.b = theano.shared(value=b_values, borrow=True)

        # convolve input feature maps with filters
        conv_out = conv.conv2d(
            input=input,
            filters=self.W,
            filter_shape=filter_shape,
            image_shape=image_shape
        )

        # downsample each feature map individually, using maxpooling
        pooled_out = downsample.max_pool_2d(
            input=conv_out,
            ds=poolsize,
            ignore_border=True
        )

        # add the bias term. Since the bias is a vector (1D array), we first
        # reshape it to a tensor of shape (1, n_filters, 1, 1). Each bias will
        # thus be broadcasted across mini-batches and feature map
        # width & height
        self.output = T.tanh(pooled_out + self.b.dimshuffle('x', 0, 'x', 'x'))

        # store parameters of this layer
        self.params = [self.W, self.b]


def evaluate_lenet5(learning_rate=0.1, n_epochs=200,
                    dataset='mnist.pkl.gz',
                    nkerns=[20, 50], batch_size=500):
    """ Demonstrates lenet on MNIST dataset

    :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: path to the dataset used for training /testing (MNIST here)

    :type nkerns: list of ints
    :param nkerns: number of kernels on each layer（两层，第一层20个卷积核，
        第二层50个卷积核）
    """

    rng = numpy.random.RandomState(23455)

    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]
    n_valid_batches = valid_set_x.get_value(borrow=True).shape[0]
    n_test_batches = test_set_x.get_value(borrow=True).shape[0]
    n_train_batches /= batch_size
    n_valid_batches /= batch_size
    n_test_batches /= batch_size

    # allocate symbolic variables for the data
    index = T.lscalar()  # index to a [mini]batch

    # start-snippet-1
    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

    ######################
    # BUILD ACTUAL MODEL #
    ######################
    print '... building the model'

    # Reshape matrix of rasterized images of shape (batch_size, 28 * 28)
    # to a 4D tensor, compatible with our LeNetConvPoolLayer
    # (28, 28) is the size of MNIST images.
    # 输入一幅图像
    layer0_input = x.reshape((batch_size, 1, 28, 28))

    # Construct the first convolutional pooling layer:
    # filtering reduces the image size to (28-5+1 , 28-5+1) = (24, 24)
    # maxpooling reduces this further to (24/2, 24/2) = (12, 12)
    # 4D output tensor is thus of shape (batch_size, nkerns[0], 12, 12)
    layer0 = LeNetConvPoolLayer(
        rng,
        input=layer0_input,
        image_shape=(batch_size, 1, 28, 28),
        filter_shape=(nkerns[0], 1, 5, 5),
        poolsize=(2, 2)
    )

    # Construct the second convolutional pooling layer
    # filtering reduces the image size to (12-5+1, 12-5+1) = (8, 8)
    # maxpooling reduces this further to (8/2, 8/2) = (4, 4)
    # 4D output tensor is thus of shape (batch_size, nkerns[1], 4, 4)
    # 由于第0层有nkerns[0]个卷积核，所以输出了nkerns[0]幅特征图
    # 第1层的输入就是第0层的输出
    layer1 = LeNetConvPoolLayer(
        rng,
        input=layer0.output,
        image_shape=(batch_size, nkerns[0], 12, 12),
        filter_shape=(nkerns[1], nkerns[0], 5, 5),
        poolsize=(2, 2)
    )

    # the HiddenLayer being fully-connected, it operates on 2D matrices of
    # shape (batch_size, num_pixels) (i.e matrix of rasterized images).
    # This will generate a matrix of shape (batch_size, nkerns[1] * 4 * 4),
    # or (500, 50 * 4 * 4) = (500, 800) with the default values.
    layer2_input = layer1.output.flatten(2)

    # construct a fully-connected sigmoidal layer
    layer2 = HiddenLayer(
        rng,
        input=layer2_input,
        n_in=nkerns[1] * 4 * 4,
        n_out=500,
        activation=T.tanh
    )

    # classify the values of the fully-connected sigmoidal layer
    layer3 = LogisticRegression(input=layer2.output, n_in=500, n_out=10)

    # the cost we minimize during training is the NLL of the model
    cost = layer3.negative_log_likelihood(y)

    # create a function to compute the mistakes that are made by the model
    test_model = theano.function(
        [index],
        layer3.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(
        [index],
        layer3.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]
        }
    )

    # create a list of all model parameters to be fit by gradient descent
    params = layer3.params + layer2.params + layer1.params + layer0.params

    # create a list of gradients for all model parameters
    grads = T.grad(cost, params)

    # train_model is a function that updates the model parameters by
    # SGD Since this model has many parameters, it would be tedious to
    # manually create an update rule for each model parameter. We thus
    # create the updates list by automatically looping over all
    # (params[i], grads[i]) pairs.
    updates = [
        (param_i, param_i - learning_rate * grad_i)
        for param_i, grad_i in zip(params, grads)
    ]

    train_model = theano.function(
        [index],
        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-1

    ###############
    # 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

    while (epoch < n_epochs) and (not done_looping):
        epoch = epoch + 1
        for minibatch_index in xrange(n_train_batches):

            iter = (epoch - 1) * n_train_batches + minibatch_index

            if iter % 100 == 0:
                print 'training @ iter = ', iter
            cost_ij = train_model(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)

                    # save best validation score and iteration number
                    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.))

            if patience <= iter:
                done_looping = True
                break

    end_time = time.clock()
    print('Optimization complete.')
    print('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__':
    evaluate_lenet5()


def experiment(state, channel):
    evaluate_lenet5(state.learning_rate, dataset=state.dataset)

在GeForce GT 720M GPU上运行170多分钟

9.训练技巧

• 滤波器的数量：计算一个卷积滤波器要比训练传统的MLPs花费更多的时间！由于特征图的尺寸随着深度不断减小，所以在靠近输出层的时候，滤波器（卷积核）的数量通常比较少。为了保留输入层的信息，激活单元的数量在层数增加的时候要保证不能减少。
• 滤波器尺寸：滤波器尺寸通常依赖于数据集。在Minist数据集上最好的尺寸是5*5，通常的自然图像较好的是12*12或者15*15
• 池化尺寸：典型的值就是2*2，对于很大的输入，可以在较低的层上使用4*4，但是记住，这将会使得信号的维度降低为原来的1/16,可能会损失太多的信息

学习资料来源：http://deeplearning.net/tutorial/lenet.html#lenet