随笔-252  评论-2258  文章-0 

Deep learning:四十五(maxout简单理解)

 

  maxout出现在ICML2013上,作者Goodfellow将maxout和dropout结合后,号称在MNIST, CIFAR-10, CIFAR-100, SVHN这4个数据上都取得了start-of-art的识别率。

  从论文中可以看出,maxout其实一种激发函数形式。通常情况下,如果激发函数采用sigmoid函数的话,在前向传播过程中,隐含层节点的输出表达式为:

   

  其中W一般是2维的,这里表示取出的是第i列,下标i前的省略号表示对应第i列中的所有行。但如果是maxout激发函数,则其隐含层节点的输出表达式为:

    

  

  这里的W是3维的,尺寸为d*m*k,其中d表示输入层节点的个数,m表示隐含层节点的个数,k表示每个隐含层节点对应了k个”隐隐含层”节点,这k个”隐隐含层”节点都是线性输出的,而maxout的每个节点就是取这k个”隐隐含层”节点输出值中最大的那个值。因为激发函数中有了max操作,所以整个maxout网络也是一种非线性的变换。因此当我们看到常规结构的神经网络时,如果它使用了maxout激发,则我们头脑中应该自动将这个”隐隐含层”节点加入。参考个日文的maxout ppt 中的一页ppt如下:

   

  ppt中箭头前后示意图大家应该可以明白什么是maxout激发函数了。

  maxout的拟合能力是非常强的,它可以拟合任意的的凸函数。最直观的解释就是任意的凸函数都可以由分段线性函数以任意精度拟合(学过高等数学应该能明白),而maxout又是取k个隐隐含层节点的最大值,这些”隐隐含层"节点也是线性的,所以在不同的取值范围下,最大值也可以看做是分段线性的(分段的个数与k值有关)。论文中的图1如下(它表达的意思就是可以拟合任意凸函数,当然也包括了ReLU了):

   

  作者从数学的角度上也证明了这个结论,即只需2个maxout节点就可以拟合任意的凸函数了(相减),前提是”隐隐含层”节点的个数可以任意多,如下图所示:

   

  下面来看下maxout源码,看其激发函数表达式是否符合我们的理解。找到库目录下的pylearn2/models/maxout.py文件,选择不带卷积的Maxout类,主要是其前向传播函数fprop():

  def fprop(self, state_below): #前向传播,对linear分组进行max-pooling操作
                                                                                                                                        
      self.input_space.validate(state_below)
                                                                                                                                        
      if self.requires_reformat:
          if not isinstance(state_below, tuple):
              for sb in get_debug_values(state_below):
                  if sb.shape[0] != self.dbm.batch_size:
                      raise ValueError("self.dbm.batch_size is %d but got shape of %d" % (self.dbm.batch_size, sb.shape[0]))
                  assert reduce(lambda x,y: x * y, sb.shape[1:]) == self.input_dim
                                                                                                                                        
          state_below = self.input_space.format_as(state_below, self.desired_space) #统一好输入数据的格式
                                                                                                                                        
      z = self.transformer.lmul(state_below) + self.b # lmul()函数返回的是 return T.dot(x, self._W)
                                                                                                                                        
      if not hasattr(self, 'randomize_pools'):
          self.randomize_pools = False
                                                                                                                                        
      if not hasattr(self, 'pool_stride'):
          self.pool_stride = self.pool_size #默认情况下是没有重叠的pooling
                                                                                                                                        
      if self.randomize_pools:
          z = T.dot(z, self.permute)
                                                                                                                                        
      if not hasattr(self, 'min_zero'):
          self.min_zero = False
                                                                                                                                        
      if self.min_zero:
          p = T.zeros_like(z) #返回一个和z同样大小的矩阵,元素值为0,元素值类型和z的类型一样
      else:
          p = None
                                                                                                                                        
      last_start = self.detector_layer_dim  - self.pool_size
      for i in xrange(self.pool_size): #xrange和reange的功能类似
          cur = z[:,i:last_start+i+1:self.pool_stride]  # L[start:end:step]是用来切片的,从[start,end)之间,每隔step取一次
          if p is None:
              p = cur
          else:
              p = T.maximum(cur, p) #将p进行迭代比较,因为每次取的是每个group里的元素,所以进行pool_size次后就可以获得每个group的最大值
                                                                                                                                        
      p.name = self.layer_name + '_p_'
                                                                                                                                        
      return p

  仔细阅读上面的源码,发现和文章中描述基本是一致的,只是多了很多细节。

  由于没有GPU,所以只用CPU 跑了个mnist的简单实验,参考:maxout下的readme文件。(需先下载mnist dataset到PYLEARN2_DATA_PATA目录下)。

  执行../../train.py minist_pi.yaml

  此时的.yaml配置文件内容如下:

!obj:pylearn2.train.Train {
    dataset: &train !obj:pylearn2.datasets.mnist.MNIST {
        which_set: 'train',
        one_hot: 1,
        start: 0,
        stop: 50000
    },
    model: !obj:pylearn2.models.mlp.MLP {
        layers: [
                 !obj:pylearn2.models.maxout.Maxout {
                     layer_name: 'h0',
                     num_units: 240,
                     num_pieces: 5,
                     irange: .005,
                     max_col_norm: 1.9365,
                 },
                 !obj:pylearn2.models.maxout.Maxout {
                     layer_name: 'h1',
                     num_units: 240,
                     num_pieces: 5,
                     irange: .005,
                     max_col_norm: 1.9365,
                 },
                 !obj:pylearn2.models.mlp.Softmax {
                     max_col_norm: 1.9365,
                     layer_name: 'y',
                     n_classes: 10,
                     irange: .005
                 }
                ],
        nvis: 784,
    },
    algorithm: !obj:pylearn2.training_algorithms.sgd.SGD {
        batch_size: 100,
        learning_rate: .1,
        learning_rule: !obj:pylearn2.training_algorithms.learning_rule.Momentum {
            init_momentum: .5,
        },
        monitoring_dataset:
            {
                'train' : *train,
                'valid' : !obj:pylearn2.datasets.mnist.MNIST {
                              which_set: 'train',
                              one_hot: 1,
                              start: 50000,
                              stop:  60000
                          },
                'test'  : !obj:pylearn2.datasets.mnist.MNIST {
                              which_set: 'test',
                              one_hot: 1,
                          }
            },
        cost: !obj:pylearn2.costs.mlp.dropout.Dropout {
            input_include_probs: { 'h0' : .8 },
            input_scales: { 'h0': 1. }
        },
        termination_criterion: !obj:pylearn2.termination_criteria.MonitorBased {
            channel_name: "valid_y_misclass",
            prop_decrease: 0.,
            N: 100
        },
        update_callbacks: !obj:pylearn2.training_algorithms.sgd.ExponentialDecay {
            decay_factor: 1.000004,
            min_lr: .000001
        }
    },
    extensions: [
        !obj:pylearn2.train_extensions.best_params.MonitorBasedSaveBest {
             channel_name: 'valid_y_misclass',
             save_path: "${PYLEARN2_TRAIN_FILE_FULL_STEM}_best.pkl"
        },
        !obj:pylearn2.training_algorithms.learning_rule.MomentumAdjustor {
            start: 1,
            saturate: 250,
            final_momentum: .7
        }
    ],
    save_path: "${PYLEARN2_TRAIN_FILE_FULL_STEM}.pkl",
    save_freq: 1
}

  跑了一个晚上才迭代了210次,被我kill掉了(笔记本还得拿到别的地方干活),这时的误差率为1.22%。估计继续跑几个小时应该会降到作者的0.94%误差率。

  其monitor监控输出结果如下:

Monitoring step:
    Epochs seen: 210
    Batches seen: 105000
    Examples seen: 10500000
    learning_rate: 0.0657047371741
    momentum: 0.667871485944
    monitor_seconds_per_epoch: 121.0
    test_h0_col_norms_max: 1.9364999
    test_h0_col_norms_mean: 1.09864382902
    test_h0_col_norms_min: 0.0935518826938
    test_h0_p_max_x.max_u: 3.97355476543
    test_h0_p_max_x.mean_u: 2.14463905251
    test_h0_p_max_x.min_u: 0.961549570265
    test_h0_p_mean_x.max_u: 0.878285389379
    test_h0_p_mean_x.mean_u: 0.131020009421
    test_h0_p_mean_x.min_u: -0.373017504665
    test_h0_p_min_x.max_u: -0.202480633479
    test_h0_p_min_x.mean_u: -1.31821964107
    test_h0_p_min_x.min_u: -2.52428183099
    test_h0_p_range_x.max_u: 5.56309069078
    test_h0_p_range_x.mean_u: 3.46285869357
    test_h0_p_range_x.min_u: 2.01775637301
    test_h0_row_norms_max: 2.67556467
    test_h0_row_norms_mean: 1.15743973628
    test_h0_row_norms_min: 0.0951322935423
    test_h1_col_norms_max: 1.12119975186
    test_h1_col_norms_mean: 0.595629304226
    test_h1_col_norms_min: 0.183531862659
    test_h1_p_max_x.max_u: 6.42944749321
    test_h1_p_max_x.mean_u: 3.74599401756
    test_h1_p_max_x.min_u: 2.03028191814
    test_h1_p_mean_x.max_u: 1.38424650414
    test_h1_p_mean_x.mean_u: 0.583690886644
    test_h1_p_mean_x.min_u: 0.0253866100292
    test_h1_p_min_x.max_u: -0.830110300894
    test_h1_p_min_x.mean_u: -1.73539242398
    test_h1_p_min_x.min_u: -3.03677525979
    test_h1_p_range_x.max_u: 8.63650239768
    test_h1_p_range_x.mean_u: 5.48138644154
    test_h1_p_range_x.min_u: 3.36428499068
    test_h1_row_norms_max: 1.95904749183
    test_h1_row_norms_mean: 1.40561339238
    test_h1_row_norms_min: 1.16953677471
    test_objective: 0.0959691806325
    test_y_col_norms_max: 1.93642459019
    test_y_col_norms_mean: 1.90996961714
    test_y_col_norms_min: 1.88659811751
    test_y_max_max_class: 1.0
    test_y_mean_max_class: 0.996910632311
    test_y_min_max_class: 0.824416386342
    test_y_misclass: 0.0114
    test_y_nll: 0.0609837733094
    test_y_row_norms_max: 0.536167736581
    test_y_row_norms_mean: 0.386866656967
    test_y_row_norms_min: 0.266996530755
    train_h0_col_norms_max: 1.9364999
    train_h0_col_norms_mean: 1.09864382902
    train_h0_col_norms_min: 0.0935518826938
    train_h0_p_max_x.max_u: 3.98463017313
    train_h0_p_max_x.mean_u: 2.16546276053
    train_h0_p_max_x.min_u: 0.986865505974
    train_h0_p_mean_x.max_u: 0.850944629066
    train_h0_p_mean_x.mean_u: 0.135825383808
    train_h0_p_mean_x.min_u: -0.354841456
    train_h0_p_min_x.max_u: -0.20750516843
    train_h0_p_min_x.mean_u: -1.32748375925
    train_h0_p_min_x.min_u: -2.49716541111
    train_h0_p_range_x.max_u: 5.61263186775
    train_h0_p_range_x.mean_u: 3.49294651978
    train_h0_p_range_x.min_u: 2.07324073262
    train_h0_row_norms_max: 2.67556467
    train_h0_row_norms_mean: 1.15743973628
    train_h0_row_norms_min: 0.0951322935423
    train_h1_col_norms_max: 1.12119975186
    train_h1_col_norms_mean: 0.595629304226
    train_h1_col_norms_min: 0.183531862659
    train_h1_p_max_x.max_u: 6.49689754011
    train_h1_p_max_x.mean_u: 3.77637040198
    train_h1_p_max_x.min_u: 2.03274038543
    train_h1_p_mean_x.max_u: 1.34966894021
    train_h1_p_mean_x.mean_u: 0.57555584546
    train_h1_p_mean_x.min_u: 0.0176827309146
    train_h1_p_min_x.max_u: -0.845786992369
    train_h1_p_min_x.mean_u: -1.74696425227
    train_h1_p_min_x.min_u: -3.05703072635
    train_h1_p_range_x.max_u: 8.73556577905
    train_h1_p_range_x.mean_u: 5.52333465425
    train_h1_p_range_x.min_u: 3.379501944
    train_h1_row_norms_max: 1.95904749183
    train_h1_row_norms_mean: 1.40561339238
    train_h1_row_norms_min: 1.16953677471
    train_objective: 0.0119584870103
    train_y_col_norms_max: 1.93642459019
    train_y_col_norms_mean: 1.90996961714
    train_y_col_norms_min: 1.88659811751
    train_y_max_max_class: 1.0
    train_y_mean_max_class: 0.999958965285
    train_y_min_max_class: 0.996295480193
    train_y_misclass: 0.0
    train_y_nll: 4.22109408992e-05
    train_y_row_norms_max: 0.536167736581
    train_y_row_norms_mean: 0.386866656967
    train_y_row_norms_min: 0.266996530755
    valid_h0_col_norms_max: 1.9364999
    valid_h0_col_norms_mean: 1.09864382902
    valid_h0_col_norms_min: 0.0935518826938
    valid_h0_p_max_x.max_u: 3.970333514
    valid_h0_p_max_x.mean_u: 2.15548653063
    valid_h0_p_max_x.min_u: 0.99228626325
    valid_h0_p_mean_x.max_u: 0.84583547397
    valid_h0_p_mean_x.mean_u: 0.143554208322
    valid_h0_p_mean_x.min_u: -0.349097300524
    valid_h0_p_min_x.max_u: -0.218285757389
    valid_h0_p_min_x.mean_u: -1.28008164111
    valid_h0_p_min_x.min_u: -2.41494612443
    valid_h0_p_range_x.max_u: 5.54136030367
    valid_h0_p_range_x.mean_u: 3.43556817173
    valid_h0_p_range_x.min_u: 2.03580165751
    valid_h0_row_norms_max: 2.67556467
    valid_h0_row_norms_mean: 1.15743973628
    valid_h0_row_norms_min: 0.0951322935423
    valid_h1_col_norms_max: 1.12119975186
    valid_h1_col_norms_mean: 0.595629304226
    valid_h1_col_norms_min: 0.183531862659
    valid_h1_p_max_x.max_u: 6.4820340666
    valid_h1_p_max_x.mean_u: 3.75160795812
    valid_h1_p_max_x.min_u: 2.00587987424
    valid_h1_p_mean_x.max_u: 1.38777592924
    valid_h1_p_mean_x.mean_u: 0.578550013139
    valid_h1_p_mean_x.min_u: 0.0232071426066
    valid_h1_p_min_x.max_u: -0.84151110053
    valid_h1_p_min_x.mean_u: -1.73734213646
    valid_h1_p_min_x.min_u: -3.09680505839
    valid_h1_p_range_x.max_u: 8.72732563235
    valid_h1_p_range_x.mean_u: 5.48895009458
    valid_h1_p_range_x.min_u: 3.32030803638
    valid_h1_row_norms_max: 1.95904749183
    valid_h1_row_norms_mean: 1.40561339238
    valid_h1_row_norms_min: 1.16953677471
    valid_objective: 0.104670540623
    valid_y_col_norms_max: 1.93642459019
    valid_y_col_norms_mean: 1.90996961714
    valid_y_col_norms_min: 1.88659811751
    valid_y_max_max_class: 1.0
    valid_y_mean_max_class: 0.99627268242
    valid_y_min_max_class: 0.767024730168
    valid_y_misclass: 0.0122
    valid_y_nll: 0.0682986195071
    valid_y_row_norms_max: 0.536167736581
    valid_y_row_norms_mean: 0.38686665696
    valid_y_row_norms_min: 0.266996530755
Saving to mnist_pi.pkl...
Saving to mnist_pi.pkl done. Time elapsed: 3.000000 seconds
Time this epoch: 0:02:08.747395

 

 

  参考资料:

  Maxout Networks.  Ian J. Goodfellow, David Warde-Farley, Mehdi Mirza, Aaron Courville, Yoshua Bengio

       一个日文的maxout ppt

       GoodFellow在ICML上关于maxout的报告。

      maxout下的readme文件。

 

 

 

 

posted on 2013-11-18 10:10 tornadomeet 阅读(...) 评论(...) 编辑 收藏

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