CNN很多概述和要点在CS231n、Neural Networks and Deep Learning中有详细阐述，这里补充Deep Learning Tutorial中的内容。本节前提是前两节的内容，因为要用到全连接层、logistic regression层等。关于Theano：掌握共享变量，下采样，conv2d，dimshuffle的应用等。
1.卷积操作
在Theano中，ConvOp是提供卷积操作的主力。ConvOp来自theano.tensor.signal.conv.conv2d，有两个参数输入[input, W]：
1）input：对应于小批量输入图像的4维张量。尺寸为[小批量尺寸，特征映射数量（滤波器数量），图像高度，图像宽度]
2）W：对应于权重W的4维张量。尺寸为[第m层滤波器数量，m-1层滤波器数量，滤波器高度，滤波器宽度]
但是下面这段代码没有使用这个函数，而是另一个theano.tensor.nnet.conv2d，后面再做解释。
# coding=utf-8 import theano from theano import tensor as T from theano.tensor.nnet import conv import numpy import numpy import pylab from PIL import Image rng = numpy.random.RandomState(23455) input = T.tensor4(name='input') #初始化4维张量类型！ w_shp = (2, 3, 9, 9) #2个滤波器，3通道，9*9滤波窗口（感受野） 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')
b_shp = (2,) b = theano.shared(numpy.asarray(rng.uniform(low=-.5, high=.5, size=b_shp),dtype=input.dtype), name ='b') conv_out = conv.conv2d(input, W) #求卷积 output = T.nnet.sigmoid(conv_out + b.dimshuffle('x', 0, 'x', 'x')) f = theano.function([input], output) #卷积操作函数 img = Image.open('3wolfmoon.jpg') #文档中给出的3狼图像（639，516，3） img = numpy.asarray(img, dtype='float64') / 256. img_ = img.transpose(2, 0, 1).reshape(1, 3, 639, 516) #图像变形为（1,3,639,516） filtered_img = f(img_) #求卷积 pylab.subplot(1, 3, 1); pylab.axis('off'); pylab.imshow(img) pylab.gray(); pylab.subplot(1, 3, 2); pylab.axis('off'); pylab.imshow(filtered_img[0, 0, :, :]) #第一滤波器结果 pylab.subplot(1, 3, 3); pylab.axis('off'); pylab.imshow(filtered_img[0, 1, :, :]) #第二滤波器结果 pylab.show()
代码结果：
由图中可以看出，随机初始化形成的滤波器经过卷积操作类似于边缘描述子
2.池化（pooling）
Cnn的一个重要步骤是池化，是一种非线性的下采样。比较重要和常见的是最大值采样。在Theano中用 theano.tensor.signal.downsample.max_pool_2d来进行。输入为N维张量（tensor）N>2。下面有一个应用例子，分别是忽略边界和不忽略边界：
from theano.tensor.signal import downsample input = T.dtensor4(’input’) maxpool_shape = (2, 2) #2*2的一个池化窗口
pool_out = downsample.max_pool_2d(input, maxpool_shape, ignore_border=True) #忽略边界的池化 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) #保留边界的池化 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, :, :]
3.完整模型：LeNet
Sparse（稀疏连接），convolutional layers（卷积层）和max-pooling（最大值池化）是LeNet家族模型的核心。虽然细节差别很大，下图展示了LeNet几何模型：
上图结构很明了，（卷积+池化）*2+全连接层（MLP），这个全连接层是很传统的一种，包含隐层+logsitic regression，这俩前两节都有介绍。现在讨论theano.tensor.nnet.conv2d和theano.tensor.signal.conv.conv.2d.前者在目前几乎所有模型中使用最多，在这个操作中，每个输出的特征映射与输入的特征映射通过2维滤波器相联系，其值为通过对应滤波器进行卷积操作的和。在原始LeNet中，输出特征映射只与输入特征映射的子集有关系。那么后者只用在信号处理中。
4.主代码
# coding=UTF-8 from __future__ import print_function import os import sys import timeit import numpy import theano import theano.tensor as T from theano.tensor.signal import pool from theano.tensor.nnet import conv2d 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)): 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:]) # 维度拉成列，每个元素都为一个像素，fan_out同理 # 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)) W_bound = numpy.sqrt(6. / (fan_in + fan_out)) self.W = theano.shared(numpy.asarray(rng.uniform(low=-W_bound, high=W_bound, size=filter_shape), dtype=theano.config.floatX),borrow=True) b_values = numpy.zeros((filter_shape[0],), dtype=theano.config.floatX) self.b = theano.shared(value=b_values, borrow=True) conv_out = conv2d( #利用滤波器进行卷积操作 input=input, filters=self.W, filter_shape=filter_shape, input_shape=image_shape ) pooled_out = pool.pool_2d( #池化：最大值池化 input=conv_out, ds=poolsize, ignore_border=True ) self.output = T.tanh(pooled_out + self.b.dimshuffle('x', 0, 'x', 'x')) #对阈值参数b维度进行调整 self.params = [self.W, self.b] #'x'看作1,0看作第零维度，这里调整后为b=(1,0维度,1,1) self.input = input #若b本身为(5,1)，则零维度为5，即b=(1,5,1,1) def evaluate_lenet5(learning_rate=0.1, n_epochs=200,dataset='mnist.pkl.gz',nkerns=[20, 50], batch_size=500): rng = numpy.random.RandomState(23455) #nkerns:两次卷积的滤波器个数本别为20，50 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] 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 index = T.lscalar() x = T.matrix('x') y = T.ivector('y') print('... building the model') layer0_input = x.reshape((batch_size, 1, 28, 28)) #mnist数据集图片尺寸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( #输入（batch_size,1,28,28）,输出（batch_size,20,12,12） rng, input=layer0_input, image_shape=(batch_size, 1, 28, 28), filter_shape=(nkerns[0], 1, 5, 5), #滤波器个数，灰度图像通道数为1，5*5的感受野 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) layer1 = LeNetConvPoolLayer( #输入（batch_size,20,12,12），输出（batch_size,1,4,4） 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) # 因为要进入全连接层，拉成一维向量即50*4*4 # construct a fully-connected sigmoidal layer layer2 = HiddenLayer( #输入50*4*4，输出500 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) #输入500，输出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( #测试模型 [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( #验证模型 [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] } ) params = layer3.params + layer2.params + layer1.params + layer0.params #参数集 grads = T.grad(cost, params) #求梯度 updates = [(param_i, param_i - learning_rate * grad_i) for param_i, grad_i in zip(params, grads)] # 参数太多，寻找更新方式太冗长，所以利用SGD更新（来自翻译） train_model = theano.function( #训练模型 [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] } ) print('... training') # early-stopping 策略 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 = timeit.default_timer() epoch = 0 done_looping = False while (epoch < n_epochs) and (not done_looping): epoch = epoch + 1 for minibatch_index in range(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 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) # 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 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.)) if patience <= iter: done_looping = True break end_time = timeit.default_timer() 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(('The code for file ' + os.path.split(__file__)[1] + ' ran for %.2fm' % ((end_time - start_time) / 60.)), file=sys.stderr) if __name__ == '__main__': evaluate_lenet5() def experiment(state, channel): evaluate_lenet5(state.learning_rate, dataset=state.dataset)