使用卷积神经网络做回归任务

Caffe应该是目前深度学习领域应用最广泛的几大框架之一了，尤其是视觉领域。绝大多数用Caffe的人，应该用的都是基于分类的网络，但有的时候也许会有基于回归的视觉应用的需要，查了一下Caffe官网，还真没有很现成的例子。这篇举个简单的小例子说明一下如何用Caffe和卷积神经网络（CNN: Convolutional Neural Networks）做基于回归的应用。

原理

最经典的CNN结构一般都是几个卷积层，后面接全连接（FC: Fully Connected）层，最后接一个Softmax层输出预测的分类概率。如果把图像的矩阵也看成是一个向量的话，CNN中无论是卷积还是FC，就是不断地把一个向量变换成另一个向量（事实上对于单个的filter/feature channel，Caffe里最基础的卷积实现就是向量和矩阵的乘法：Convolution in Caffe: a memo），最后输出就是一个把制定分类的类目数作为维度的概率向量。因为神经网络的风格算是黑盒子学习，所以很直接的想法就是把最后输出的向量的值直接拿来做回归，最后优化的目标函数不再是cross entropy等，而是直接基于实数值的误差。

EuclideanLossLayer

Caffe内置的EuclideanLossLayer就是用来解决上面提到的实值回归的一个办法。EuclideanLossLayer计算如下的误差：

egin{align} otag frac 1 {2N} sum_{i=1}^N | x^1_i - x^2_i |_2^2end{align}

所以很简单，把标注的值和网络计算出来的值放到EuclideanLossLayer比较差异就可以了。

给图像混乱程度打分的简单例子

用一个给图像混乱程度打分的简单例子来说明如何使用Caffe和EuclideanLossLayer进行回归。

生成基于Ising模型的数据

这里采用统计物理里非常经典的Ising模型的模拟来生成图片，Ising模型可能是统计物理里被人研究最多的模型之一，不过这篇不是讲物理，就略过细节，总之基于这个模型的模拟可以生成如下的图片：

图片中第一个字段是编号，第二个字段对应的分数可以大致认为是图片的有序程度，范围0~1，而这个例子要做的事情就是用一个CNN学习图片的有序程度并预测。

生成图片的Python脚本源于Monte Carlo Simulation of the Ising Model using Python，基于Metropolis算法对Ising模型的模拟，做了一些并行和随机生成图片的修改，在每次模拟的时候随机取一个时间（1e3到1e7之间）点输出到图片，代码如下：

import os
import sys
import datetime

from multiprocessing import Process

import numpy as np
from matplotlib import pyplot

LATTICE_SIZE = 100
SAMPLE_SIZE = 12000
STEP_ORDER_RANGE = [3, 7]
SAMPLE_FOLDER = 'samples'

#----------------------------------------------------------------------#
#   Check periodic boundary conditions
#----------------------------------------------------------------------#
def bc(i):
    if i+1 > LATTICE_SIZE-1:
        return 0
    if i-1 < 0:
        return LATTICE_SIZE - 1
    else:
        return i

#----------------------------------------------------------------------#
#   Calculate internal energy
#----------------------------------------------------------------------#
def energy(system, N, M):
    return -1 * system[N,M] * (system[bc(N-1), M] 
                               + system[bc(N+1), M] 
                               + system[N, bc(M-1)] 
                               + system[N, bc(M+1)])

#----------------------------------------------------------------------#
#   Build the system
#----------------------------------------------------------------------#
def build_system():
    system = np.random.random_integers(0, 1, (LATTICE_SIZE, LATTICE_SIZE))
    system[system==0] = - 1

    return system

#----------------------------------------------------------------------#
#   The Main monte carlo loop
#----------------------------------------------------------------------#
def main(T, index):

    score = np.random.random()
    order = score*(STEP_ORDER_RANGE[1]-STEP_ORDER_RANGE[0]) + STEP_ORDER_RANGE[0]
    stop = np.int(np.round(np.power(10.0, order)))
    print('Running sample: {}, stop @ {}'.format(index, stop))
    sys.stdout.flush()

    system = build_system()

    for step in range(stop):
        M = np.random.randint(0, LATTICE_SIZE)
        N = np.random.randint(0, LATTICE_SIZE)

        E = -2. * energy(system, N, M)

        if E <= 0.:
            system[N,M] *= -1
        elif np.exp(-1./T*E) > np.random.rand():
            system[N,M] *= -1

        #if step % 100000 == 0:
        #    print('.'),
        #    sys.stdout.flush()

    filename = '{}/'.format(SAMPLE_FOLDER) + '{:0>5d}'.format(index) + '_{}.jpg'.format(score)
    pyplot.imsave(filename, system, cmap='gray')
    print('Saved to {}!
'.format(filename))
    sys.stdout.flush()

#----------------------------------------------------------------------#
#   Run the menu for the monte carlo simulation
#----------------------------------------------------------------------#

def run_main(index, length):
    np.random.seed(datetime.datetime.now().microsecond)
    for i in xrange(index, index+length):
        main(0.1, i)

def run():

    cmd = 'mkdir -p {}'.format(SAMPLE_FOLDER)
    os.system(cmd)

    n_processes = 8
    length = int(SAMPLE_SIZE/n_processes)
    processes = [Process(target=run_main, args=(x, length)) for x in np.arange(n_processes)*length]

    for p in processes:
        p.start()
    
    for p in processes:
        p.join()

if __name__ == '__main__':
    run()

在这个例子中一共随机生成了12000张100x100的灰度图片，命名的规则是[编号]_[有序程度].jpg。至于有序程度为什么用0~1之间的随机数而不是模拟的时间步数，是因为虽说理论上三层神经网络就能逼近任意函数，不过具体到实际训练中还是应该对数据进行预处理，尤其是当目标函数是L2 norm的形式时，如果能保持数据分布均匀，模型的收敛性和可靠性都会提高，范围0到1之间是为了方便最后一层Sigmoid输出对比，同时也方便估算模型误差。还有一点需要注意是，因为图片本身就是模特卡罗模拟产生的，所以即使是同样的有序度的图片，其实看上去不管是主观还是客观的有序程度都是有差别的。

生成训练/验证/测试集

把Ising模拟生成的12000张图片划分为三部分：1w作为训练数据；1k作为验证集；剩下1k作为测试集。下面的Python代码用来生成这样的训练集和验证集的列表：

import os
import numpy

filename2score = lambda x: x[:x.rfind('.')].split('_')[-1]

img_files = sorted(os.listdir('samples'))

with open('train.txt', 'w') as train_txt:
    for f in img_files[:10000]:
        score = filename2score(f)
        line = 'samples/{} {}
'.format(f, score)
        train_txt.write(line)

with open('val.txt', 'w') as val_txt:
    for f in img_files[10000:11000]:
        score = filename2score(f)
        line = 'samples/{} {}
'.format(f, score)
        val_txt.write(line)

with open('test.txt', 'w') as test_txt:
    for f in img_files[11000:]:
        line = 'samples/{}
'.format(f)
        test_txt.write(line)

生成HDF5文件

lmdb虽然又快又省空间，可是Caffe默认的生成lmdb的工具(convert_imageset)不支持浮点类型的数据，虽然caffe.proto里Datum的定义似乎是支持的，不过相应的代码改动还是比较麻烦。相比起来HDF又慢又占空间，但简单好用，如果不是海量数据，还是个不错的选择，这里用HDF来存储用于回归训练和验证的数据，下面是一个生成HDF文件和供Caffe读取文件列表的脚本：

import sys
import numpy
from matplotlib import pyplot
import h5py

IMAGE_SIZE = (100, 100)
MEAN_VALUE = 128

filename = sys.argv[1]
setname, ext = filename.split('.')

with open(filename, 'r') as f:
    lines = f.readlines()

numpy.random.shuffle(lines)

sample_size = len(lines)
imgs = numpy.zeros((sample_size, 1,) + IMAGE_SIZE, dtype=numpy.float32)
scores = numpy.zeros(sample_size, dtype=numpy.float32)

h5_filename = '{}.h5'.format(setname)
with h5py.File(h5_filename, 'w') as h:
    for i, line in enumerate(lines):
        image_name, score = line[:-1].split()
        img = pyplot.imread(image_name)[:, :, 0].astype(numpy.float32)
        img = img.reshape((1, )+img.shape)
        img -= MEAN_VALUE
        imgs[i] = img
        scores[i] = float(score)
        if (i+1) % 1000 == 0:
            print('processed {} images!'.format(i+1))
    h.create_dataset('data', data=imgs)
    h.create_dataset('score', data=scores)

with open('{}_h5.txt'.format(setname), 'w') as f:
    f.write(h5_filename)

需要注意的是Caffe中HDF的DataLayer不支持transform，所以数据存储前就提前进行了减去均值的步骤。保存为gen_hdf.py，依次运行命令生成训练集和验证集：

python gen_hdf.py train.txt
python gen_hdf.py val.txt

训练

用一个简单的小网络训练这个基于回归的模型：

网络结构的train_val.prototxt如下：

name: "RegressionExample"
layer {
  name: "data"
  type: "HDF5Data"
  top: "data"
  top: "score"
  include {
    phase: TRAIN
  }
  hdf5_data_param {
    source: "train_h5.txt"
    batch_size: 64
  }
}
layer {
  name: "data"
  type: "HDF5Data"
  top: "data"
  top: "score"
  include {
    phase: TEST
  }
  hdf5_data_param {
    source: "val_h5.txt"
    batch_size: 64
  }
}
layer {
  name: "conv1"
  type: "Convolution"
  bottom: "data"
  top: "conv1"
  param {
    lr_mult: 1
    decay_mult: 1
  }
  param {
    lr_mult: 1
    decay_mult: 0
  }
  convolution_param {
    num_output: 96
    kernel_size: 5
    stride: 2
    weight_filler {
      type: "gaussian"
      std: 0.01
    }
    bias_filler {
      type: "constant"
      value: 0
    }
  }
}
layer {
  name: "relu1"
  type: "ReLU"
  bottom: "conv1"
  top: "conv1"
}
layer {
  name: "pool1"
  type: "Pooling"
  bottom: "conv1"
  top: "pool1"
  pooling_param {
    pool: MAX
    kernel_size: 3
    stride: 2
  }
}
layer {
  name: "conv2"
  type: "Convolution"
  bottom: "pool1"
  top: "conv2"
  param {
    lr_mult: 1
    decay_mult: 1
  }
  param {
    lr_mult: 1
    decay_mult: 0
  }
  convolution_param {
    num_output: 96
    pad: 2
    kernel_size: 3
    weight_filler {
      type: "gaussian"
      std: 0.01
    }
    bias_filler {
      type: "constant"
      value: 0
    }
  }
}
layer {
  name: "relu2"
  type: "ReLU"
  bottom: "conv2"
  top: "conv2"
}
layer {
  name: "pool2"
  type: "Pooling"
  bottom: "conv2"
  top: "pool2"
  pooling_param {
    pool: MAX
    kernel_size: 3
    stride: 2
  }
}
layer {
  name: "conv3"
  type: "Convolution"
  bottom: "pool2"
  top: "conv3"
  param {
    lr_mult: 1
    decay_mult: 1
  }
  param {
    lr_mult: 1
    decay_mult: 0
  }
  convolution_param {
    num_output: 128
    pad: 1
    kernel_size: 3
    weight_filler {
      type: "gaussian"
      std: 0.01
    }
    bias_filler {
      type: "constant"
      value: 0
    }
  }
}
layer {
  name: "relu3"
  type: "ReLU"
  bottom: "conv3"
  top: "conv3"
}
layer {
  name: "pool3"
  type: "Pooling"
  bottom: "conv3"
  top: "pool3"
  pooling_param {
    pool: MAX
    kernel_size: 3
    stride: 2
  }
}
layer {
  name: "fc4"
  type: "InnerProduct"
  bottom: "pool3"
  top: "fc4"
  param {
    lr_mult: 1
    decay_mult: 1
  }
  param {
    lr_mult: 1
    decay_mult: 0
  }
  inner_product_param {
    num_output: 192
    weight_filler {
      type: "gaussian"
      std: 0.005
    }
    bias_filler {
      type: "constant"
      value: 0
    }
  }
}
layer {
  name: "relu4"
  type: "ReLU"
  bottom: "fc4"
  top: "fc4"
}
layer {
  name: "drop4"
  type: "Dropout"
  bottom: "fc4"
  top: "fc4"
  dropout_param {
    dropout_ratio: 0.35
  }
}
layer {
  name: "fc5"
  type: "InnerProduct"
  bottom: "fc4"
  top: "fc5"
  param {
    lr_mult: 1
    decay_mult: 1
  }
  param {
    lr_mult: 1
    decay_mult: 0
  }
  inner_product_param {
    num_output: 1
    weight_filler {
      type: "gaussian"
      std: 0.005
    }
    bias_filler {
      type: "constant"
      value: 0
    }
  }
}
layer {
  name: "sigmoid5"
  type: "Sigmoid"
  bottom: "fc5"
  top: "pred"
}
layer {
  name: "loss"
  type: "EuclideanLoss"
  bottom: "pred"
  bottom: "score"
  top: "loss"
}

其中回归部分由EuclideanLossLayer中???较最后一层的输出和train.txt/val.txt中的分数差并作为目标函数实现。需要提一句的是基于实数值的回归问题，对于方差这种目标函数，SGD的性能和稳定性一般来说都不是很好，Caffe文档里也有提到过这点。不过具体到Caffe中，能用就行。。solver.prototxt如下：

net: "./train_val.prototxt"
test_iter: 2000
test_interval: 500
base_lr: 0.01
lr_policy: "step"
gamma: 0.1
stepsize: 50000
display: 50
max_iter: 10000
momentum: 0.85
weight_decay: 0.0005
snapshot: 1000
snapshot_prefix: "./example_ising"
solver_mode: GPU
type: "Nesterov"

然后训练：

/path/to/caffe/build/tools/caffe train -solver solver.prototxt

测试

随便训了10000个iteration，反正是收敛了

把train_val.prototxt的两个data layer替换成input_shape，然后去掉最后一层EuclideanLoss就可以了，input_shape定义如下：

input: "data"
input_shape {
  dim: 1
  dim: 1
  dim: 100
  dim: 100
}

改好后另存为deploy.prototxt，然后把训好的模型拿来在测试集上做测试，pycaffe提供了非常方便的接口，用下面脚本输出一个文件列表里所有文件的预测结果：

import sys
import numpy
sys.path.append('/opt/caffe/python')
import caffe

WEIGHTS_FILE = 'example_ising_iter_10000.caffemodel'
DEPLOY_FILE = 'deploy.prototxt'
IMAGE_SIZE = (100, 100)
MEAN_VALUE = 128

caffe.set_mode_cpu()
net = caffe.Net(DEPLOY_FILE, WEIGHTS_FILE, caffe.TEST)
net.blobs['data'].reshape(1, 1, *IMAGE_SIZE)

transformer = caffe.io.Transformer({'data': net.blobs['data'].data.shape})
transformer.set_transpose('data', (2,0,1))
transformer.set_mean('data', numpy.array([MEAN_VALUE]))
transformer.set_raw_scale('data', 255)

image_list = sys.argv[1]

with open(image_list, 'r') as f:
    for line in f.readlines():
        filename = line[:-1]
        image = caffe.io.load_image(filename, False)
        transformed_image = transformer.preprocess('data', image)
        net.blobs['data'].data[...] = transformed_image

        output = net.forward()
        score = output['pred'][0][0]

        print('The predicted score for {} is {}'.format(filename, score))

对test.txt执行后，前20个文件的结果：

The predicted score for samples/11000_0.30434289374.jpg is 0.296356916428
The predicted score for samples/11001_0.865486910668.jpg is 0.823452055454
The predicted score for samples/11002_0.566940975024.jpg is 0.566108822823
The predicted score for samples/11003_0.447787648857.jpg is 0.443993896246
The predicted score for samples/11004_0.688095649282.jpg is 0.714970111847
The predicted score for samples/11005_0.0834013155212.jpg is 0.0675165131688
The predicted score for samples/11006_0.421206628337.jpg is 0.419887691736
The predicted score for samples/11007_0.579389741639.jpg is 0.58779758215
The predicted score for samples/11008_0.428772434501.jpg is 0.422569811344
The predicted score for samples/11009_0.188864264594.jpg is 0.18296033144
The predicted score for samples/11010_0.328103100948.jpg is 0.325099766254
The predicted score for samples/11011_0.131306426901.jpg is 0.119059860706
The predicted score for samples/11012_0.627027363247.jpg is 0.622474730015
The predicted score for samples/11013_0.0857273267817.jpg is 0.0735778361559
The predicted score for samples/11014_0.870007364446.jpg is 0.883266746998
The predicted score for samples/11015_0.0515036691772.jpg is 0.0575885437429
The predicted score for samples/11016_0.799989222638.jpg is 0.750781834126
The predicted score for samples/11017_0.22049410733.jpg is 0.208014890552
The predicted score for samples/11018_0.882973794598.jpg is 0.891137182713
The predicted score for samples/11019_0.686353385772.jpg is 0.671325206757
The predicted score for samples/11020_0.385639405472.jpg is 0.385150641203

看上去还不错，挑几张看看：

再输出第一层的卷积核看看：

可以看到第一层的卷积核成功学到了高频和低频的成分，这也是这个例子中判断有序程度的关键，其实就是高频的图像就混乱，低频的就相对有序一些。Ising的自旋图虽然都是二值的，不过学出来的模型也可以随便拿一些别的图片试试：

嗯。。定性看还是差不多的。