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torch.nn.BatchNorm1d和torch.nn.BatchNorm2d

作者：羊村懒王 | 2024-05-22 04:41:01

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torch.nn.batchnorm1d

在机器学习中，进行模型训练之前，需对数据做归一化处理，使其分布一致。在深度神经网络训练过程中，通常一次训练一个batch，而非全体数据。每个batch具有不同的分布，这样产生了internal covarivate shift问题——在训练过程中，数据分布会发生变化，对下一层网络的学习带来困难。Batch Normalization强行将数据拉回到均值为0，方差为1的正太分布上，一方面使得数据分布一致，另一方面避免梯度消失。可以加快网络训练的收敛速度。
参考：

 “Batch normalization: Accelerating deep network training by reducing internal covariate shift.” 

PyTorch踩坑指南（1）nn.BatchNorm2d()函数_白水煮蝎子的博客-CSDN博客

<1>torch.nn.BatchNorm1d

CLASS torch.nn.BatchNorm1d(num_features, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True, device=None, dtype=None)

Parameters:

  • num_features (int) – number of features or channels C of the input.样本的特征维数，一般输入数据的形式是矩阵$batch \times C$。

  • eps (float) – a value added to the denominator for numerical stability. Default: 1e-5.为避免归一化时分母为0。

  • momentum (float) – the value used for the running_mean and running_var computation. Can be set to None for cumulative moving average (i.e. simple average). Default: 0.1 用来计算running_mean和running_var的一个量。

  • affine (bool) – a boolean value that when set to True, this module has learnable affine parameters. Default: True.用于设置是否具有可学习的仿射参数$\gamma $和$\beta $，$\gamma $的初始值为1，$\beta $的初始值为0。

  • track_running_stats (bool) – a boolean value that when set to True, this module tracks the running mean and variance, and when set to False, this module does not track such statistics, and initializes statistics buffers running_mean and running_var as None. When these buffers are None, this module always uses batch statistics. in both training and eval modes. Default: True.用于设置是否更新统计值均值与方差，如果设置为True，则running_mean 的初始值为0，running_var 的初始值为1，如果设置为False，则初始化running_mean 和running_var 为None，如果为None则在训练和测试的过程中均使用测试数据batch 的统计值进行数据归一化。

Shape:

  • Input: (N, C), where N is the batch size, Cis the number of features or channels.输入数据的形式是矩阵$N \times C$。

  • Output: (N, C) (same shape as input).

需要注意的是：仿射参数$\gamma $和$\beta $是在反向传播学习得到，running_mean 和running_var是在正向传播中统计得到。

参考：BatchNorm1d — PyTorch 2.0 documentation

nn.BatchNorm1d_harry_tea的博客-CSDN博客

1.模型均值和方差的更新机制、数据归一化机制

需要注意的是 track_running_stats 的设置只在创建BatchNorm时有效，不在创建BatchNorm时设置不起效果，如下面案例：

  1. import torch
  2. m = torch.nn.BatchNorm1d(8,momentum=0.1, affine=True, track_running_stats = True) #track_running_stats只在初始化时设置有效
  3. m.track_running_stats = False
  4. print('running_mean:',m.running_mean) #初始值
  5. print('running_var:',m.running_var )
  6. print('track_running_stats:',m.track_running_stats )
  7. m2 = torch.nn.BatchNorm1d(8,momentum=0.1, affine=True, track_running_stats = False) 
  8. print('running_mean:',m2.running_mean) #初始值
  9. print('running_var:',m2.running_var )
  1. running_mean: tensor([0., 0., 0., 0., 0., 0., 0., 0.])
  2. running_var: tensor([1., 1., 1., 1., 1., 1., 1., 1.])
  3. track_running_stats: False
  4. running_mean: None
  5. running_var: None

running_mean 和running_var 模型参数是否需要更新，需要结合参数trainning和track_running_states来看，归一化的机制也因这两种参数的不同而不同。

(1)trainning = True，track_running_states = True：模型处于训练阶段，每作一次归一化，模型都需要更新参数running_mean和running_var，即跟踪每个batch数据的均值和方差。

参数更新方法：${x_{new}} = (1 - momentum)*{x_{old}} + momentum*{x_{obser}}$

其中，${x_{old}} $为模型的均值或方差，${x_{obser}} $为当前观测数据batch的均值或方差，${x_{new}} $为更新后的均值或方差，$momentum $为更新参数。

观测数据batch归一化方法：$y = {​{x - E[x]} \over {\sqrt {Var[x] + \varepsilon } }}*\gamma + \beta $

其中，$E[x] $为观测数据batch的均值，$Var[x] $为观测数据batch的方差，$y $为归一化之后的某个通道的数据，即归一化是用当前batch的均值和方差做归一化。

注意方差的无偏估计：${\sigma ^2} = {​{\sum\limits_{i = 0}^{N - 1} {​{​{({x_i} - E(x))}^2}} } \mathord{\left/ {\vphantom {​{\sum\limits_{i = 0}^{N - 1} {​{​{({x_i} - E(x))}^2}} } {(N - 1)}}} \right. \kern-\nulldelimiterspace} {(N - 1)}}$

方差的有偏估计：${\sigma ^2} = {​{\sum\limits_{i = 0}^{N - 1} {​{​{({x_i} - E(x))}^2}} } \mathord{\left/ {\vphantom {​{\sum\limits_{i = 0}^{N - 1} {​{​{({x_i} - E(x))}^2}} } {(N - 1)}}} \right. \kern-\nulldelimiterspace} {N }}$，

需要注意的是：running_mean和running_var更新用的无偏的方差，数据归一化用的是有偏的方差。当数值$N$较大时，有偏估计和无偏估计基本一致。

  1. import torch
  2. m = torch.nn.BatchNorm1d(8,momentum=0.1, affine=True, track_running_stats = True) #track_running_stats只在初始化时设置有效
  3. print('running_mean:',m.running_mean) #初始值
  4. print('running_var:',m.running_var )
  5. print('weight:',m.weight)
  6. print('bias:',m.bias)
  7.  
  8. input = torch.randn(5, 8)
  9. print('input:',input)
  10. print('input[...,0]:',input[...,0])#第一列数据
  11.  
  12. obser_mean = torch.Tensor([input[...,i].mean() for i in range(8)])# 输入数据的均值
  13. obser_var_unbiased = torch.Tensor([input[...,i].var() for i in range(8)])# 方差 无偏估计，相当于torch.var(input, dim=0, unbiased=True)
  14. obser_var_biased = torch.Tensor([input[...,i].var(unbiased=False) for i in range(8)])# 方差 有偏估计，相当于torch.var(input, dim=0, unbiased=False) 
  15. print('obser_mean:',obser_mean)
  16. print('obser_var_unbiased:',obser_var_unbiased)
  17. print('obser_var_biased:',obser_var_biased)
  18.  
  19. obser_running_mean = (1-m.momentum)*m.running_mean + m.momentum*obser_mean
  20. obser_running_var = (1-m.momentum)*m.running_var + m.momentum*obser_var_unbiased
  21. output = m(input)
  22. output_obser = (input[...,0] - obser_mean[0])/(pow(obser_var_biased[0] + m.eps,0.5))#归一化
  23. print('obser_running_mean:',obser_running_mean)
  24. print('obser_running_var:',obser_running_var)
  25. print('running_mean:',m.running_mean)
  26. print('running_var:',m.running_var )
  27. print('output[...,0]:',output[...,0])#归一化数据
  28. print('output_obser:',output_obser)
  1. running_mean: tensor([0., 0., 0., 0., 0., 0., 0., 0.])
  2. running_var: tensor([1., 1., 1., 1., 1., 1., 1., 1.])
  3. weight: Parameter containing:
  4. tensor([1., 1., 1., 1., 1., 1., 1., 1.], requires_grad=True)
  5. bias: Parameter containing:
  6. tensor([0., 0., 0., 0., 0., 0., 0., 0.], requires_grad=True)
  7. input: tensor([[ 0.4901, -0.1794,  1.1301, -0.1901, -0.7794, -0.2863, -0.9673,  1.5712],
  8.         [ 0.7150, -0.6555,  0.1724,  1.8487,  0.3064, -0.0863,  1.3970,  0.3117],
  9.         [-1.4870, -1.0768, -0.8371,  1.7132,  0.9250,  0.6004, -0.2488,  0.8714],
  10.         [-0.7459,  0.3344, -1.1203,  1.7061,  0.6755, -0.2490,  1.4969,  0.6247],
  11.         [-0.2600,  2.0536,  0.5194, -1.4121, -1.3856, -0.2249,  0.0729, -0.4737]])
  12. input[...,0]: tensor([ 0.4901,  0.7150, -1.4870, -0.7459, -0.2600])
  13. obser_mean: tensor([-0.2576,  0.0953, -0.0271,  0.7332, -0.0516, -0.0492,  0.3501,  0.5810])
  14. obser_var_unbiased: tensor([0.8138, 1.4763, 0.8822, 2.1516, 0.9799, 0.1376, 1.1456, 0.5629])
  15. obser_var_biased: tensor([0.6510, 1.1810, 0.7058, 1.7213, 0.7839, 0.1101, 0.9165, 0.4503])
  16. obser_running_mean: tensor([-0.0258,  0.0095, -0.0027,  0.0733, -0.0052, -0.0049,  0.0350,  0.0581])
  17. obser_running_var: tensor([0.9814, 1.0476, 0.9882, 1.1152, 0.9980, 0.9138, 1.0146, 0.9563])
  18. running_mean: tensor([-0.0258,  0.0095, -0.0027,  0.0733, -0.0052, -0.0049,  0.0350,  0.0581])
  19. running_var: tensor([0.9814, 1.0476, 0.9882, 1.1152, 0.9980, 0.9138, 1.0146, 0.9563])
  20. output[...,0]: tensor([ 0.9267,  1.2054, -1.5237, -0.6053, -0.0031],
  21.        grad_fn=<SelectBackward0>)
  22. output_obser: tensor([ 0.9267,  1.2054, -1.5237, -0.6053, -0.0031])

(2)trainning = False，track_running_states = True：模型处于测试阶段，利用模型存储的均值和方差作归一化，但不更新模型的均值和方差。

  1. # 测试阶段
  2. m.eval()
  3. print(m.training)
  4. print(m.track_running_stats)
  5. input = torch.randn(5, 8)
  6.  
  7. obser_mean = torch.mean(input, dim=0)# 输入数据的均值
  8. obser_var_biased = torch.var(input, dim=0, unbiased=False) # 方差 有偏估计
  9. print('obser_mean:',obser_mean)
  10. print('obser_var_biased:',obser_var_biased)
  11. print('running_mean:',m.running_mean)
  12. print('running_var:',m.running_var )
  13.  
  14. output = m(input)
  15. output_obser = (input[...,0] - obser_mean[0])/(pow(obser_var_biased[0] + m.eps,0.5))#归一化
  16. output2_obser = (input[...,0] - m.running_mean[0])/(pow(m.running_var[0] + m.eps,0.5))
  17. print('output[...,0]:',output[...,0])
  18. print('output_obser:',output_obser)
  19. print('output2_obser:',output2_obser)
  1. False
  2. True
  3. obser_mean: tensor([-0.1944, -0.0978, -0.2910,  0.1413, -0.3192,  0.2840,  0.2374,  0.0797])
  4. obser_var_biased: tensor([2.0099, 0.1910, 0.8365, 0.2942, 1.6348, 1.7859, 1.5220, 0.6940])
  5. running_mean: tensor([ 0.0830,  0.0147,  0.0465,  0.0591,  0.0311,  0.0579, -0.0096,  0.0033])
  6. running_var: tensor([1.0621, 0.9541, 0.9964, 1.1851, 0.9764, 1.0526, 1.0279, 0.9809])
  7. output[...,0]: tensor([ 0.4747, -2.2079, -1.2251,  1.7868, -0.1744],
  8.        grad_fn=<SelectBackward0>)
  9. output_obser: tensor([ 0.5407, -1.4093, -0.6949,  1.4946,  0.0689])
  10. output2_obser: tensor([ 0.4747, -2.2079, -1.2251,  1.7868, -0.1744])

(3)trainning = True或False，track_running_states = False：模型无论处于训练或测试阶段，都是利用当前batch的均值和方差做归一化，且不更新模型的均值和方差。

  1. import torch
  2. m = torch.nn.BatchNorm1d(8,momentum=0.1, affine=True, track_running_stats = False) #track_running_stats只在初始化时设置有效
  3. print('running_mean:',m.running_mean) #初始值
  4. print('running_var:',m.running_var )
  5.  
  6. input = torch.randn(5, 8)
  7. obser_mean = torch.Tensor([input[...,i].mean() for i in range(8)])# 输入数据的均值
  8. obser_var_biased = torch.Tensor([input[...,i].var(unbiased=False) for i in range(8)])# 方差 有偏估计
  9. print('obser_mean:',obser_mean)
  10. print('obser_var_biased:',obser_var_biased)
  11. output = m(input)
  12. output_obser = (input[...,0] - obser_mean[0])/(pow(obser_var_biased[0] + m.eps,0.5))#归一化
  13. print('running_mean:',m.running_mean)
  14. print('running_var:',m.running_var )
  15. print('output[...,0]:',output[...,0])#归一化数据
  16. print('output_obser:',output_obser)
  17.  
  18.  
  19. # 测试阶段
  20. m.eval()
  21. print(m.training)
  22. print(m.track_running_stats)
  23. input = torch.randn(5, 8)
  24.  
  25. obser_mean = torch.mean(input, dim=0)# 输入数据的均值
  26. obser_var_biased = torch.var(input, dim=0, unbiased=False) # 方差 有偏估计
  27. print('obser_mean:',obser_mean)
  28. print('obser_var_biased:',obser_var_biased)
  29. print('running_mean:',m.running_mean)
  30. print('running_var:',m.running_var )
  31.  
  32. output = m(input)
  33. output_obser = (input[...,0] - obser_mean[0])/(pow(obser_var_biased[0] + m.eps,0.5))#归一化
  34. print('output[...,0]:',output[...,0])
  35. print('output_obser:',output_obser)
  1. running_mean: None
  2. running_var: None
  3. obser_mean: tensor([-0.4277,  0.2008, -0.3871,  0.4741,  0.5016,  0.6817, -0.2613,  0.0763])
  4. obser_var_biased: tensor([0.3961, 0.7895, 1.1211, 0.2614, 0.2954, 0.4563, 1.8461, 0.7862])
  5. running_mean: None
  6. running_var: None
  7. output[...,0]: tensor([ 0.6016,  1.0995, -1.0294,  0.6994, -1.3712],
  8.        grad_fn=<SelectBackward0>)
  9. output_obser: tensor([ 0.6016,  1.0995, -1.0294,  0.6994, -1.3712])
  10. False
  11. False
  12. obser_mean: tensor([-0.7911, -0.0979,  0.5710, -0.8198,  0.3552, -0.0772,  0.7881,  0.7573])
  13. obser_var_biased: tensor([2.0702, 1.2274, 1.2483, 0.5527, 0.3471, 0.2689, 1.0752, 0.7770])
  14. running_mean: None
  15. running_var: None
  16. output[...,0]: tensor([ 0.1504,  0.6443, -0.4143,  1.2792, -1.6596],
  17.        grad_fn=<SelectBackward0>)
  18. output_obser: tensor([ 0.1504,  0.6443, -0.4143,  1.2792, -1.6596])

<2>torch.nn.BatchNorm2d

CLASS torch.nn.BatchNorm2d(num_features, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True, device=None, dtype=None)

Parameters:

  • num_features (int) – Cfrom an expected input of size (N, C, H, W)

Shape:

  • Input: (N, C, H, W)

  • Output: (N, C, H, W) (same shape as input)

BatchNorm2d和BatchNorm1d不同的地方在于输入数据的不同。在计算观测数据batch的均值和方差的时候是对batch的同一通道上的所有元素进行统计计算，假设输入数据为$(N, C, H, W)$，则有：

每个通道的均值：${\mu _c} = {​{\sum\limits_{n = 0}^{N - 1} {\sum\limits_{h = 0}^{H - 1} {\sum\limits_{w = 0}^{W - 1} {x[n,c,w,h]} } } } \over {N \times H \times W}},c = 0, \cdots ,C - 1$。

每个通道的方差：$\sigma _c^2 = {​{\sum\limits_{n = 0}^{N - 1} {\sum\limits_{h = 0}^{H - 1} {\sum\limits_{w = 0}^{W - 1} {​{​{(x[n,c,w,h] - {\mu _c})}^2}} } } } \over {N \times H \times W}},c = 0, \cdots ,C - 1$。

对通道$c$的每个元素${x[n,c,w,h]}$作归一化：$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over x} = {​{x - {\mu _c}} \over {\sqrt {\sigma _c^2 + \varepsilon } }}*\gamma + \beta $。

BatchNorm2d的模型均值和方差的更新机制，以及数据归一化机制跟BatchNorm1d一样。

  1. import torch
  2. m = torch.nn.BatchNorm2d(3, eps=0, momentum=0.5, affine=True, track_running_stats=True)
  3. print('running_mean:',m.running_mean) #初始值
  4. print('running_var:',m.running_var )
  5. print('weight:',m.weight)
  6. print('bias:',m.bias)
  7.  
  8. input = torch.randn(1, 3, 5, 5)
  9. print('input[0][0]:',input[0][0])#第一个batch，第一个通道
  10.  
  11. obser_mean = torch.Tensor([input[0][i].mean() for i in range(3)])# 输入数据的均值
  12. obser_var_unbiased = torch.Tensor([input[0][i].var() for i in range(3)])# 方差 无偏估计，相当于torch.var(input[0][0], unbiased=True)
  13. obser_var_biased = torch.Tensor([input[0][i].var(unbiased=False) for i in range(3)])# 方差 有偏估计，相当于torch.var(input[0][0], unbiased=False)
  14. print('obser_mean:',obser_mean)
  15. print('obser_var_unbiased:',obser_var_unbiased)
  16. print('obser_var_biased:',obser_var_biased)
  17. obser_running_mean = (1-m.momentum)*m.running_mean + m.momentum*obser_mean
  18. obser_running_var = (1-m.momentum)*m.running_var + m.momentum*obser_var_unbiased
  19. output = m(input)
  20. print('obser_running_mean:',obser_running_mean)
  21. print('obser_running_var:',obser_running_var)
  22. print('running_mean:',m.running_mean)
  23. print('running_var:',m.running_var )
  24. output_obser = (input[0][0] - obser_mean[0])/(pow(obser_var_biased[0] + m.eps,0.5))#归一化
  25. print('output[0][0]:',output[0][0])#归一化数据
  26. print('output_obser:',output_obser)
  1. running_mean: tensor([0., 0., 0.])
  2. running_var: tensor([1., 1., 1.])
  3. weight: Parameter containing:
  4. tensor([1., 1., 1.], requires_grad=True)
  5. bias: Parameter containing:
  6. tensor([0., 0., 0.], requires_grad=True)
  7. input[0][0]: tensor([[-0.3550, -1.1596, -0.4947, -0.8188, -0.1722],
  8.         [-1.3371,  0.9375,  0.5564,  2.3561, -0.5711],
  9.         [ 0.3932,  2.6657,  0.3440, -0.9300,  0.1791],
  10.         [-1.0307,  0.2115,  0.4953,  1.8088,  0.0496],
  11.         [-1.0584,  0.4566, -0.1415,  1.2106,  0.4498]])
  12. obser_mean: tensor([ 0.1618,  0.2137, -0.0836])
  13. obser_var_unbiased: tensor([1.1056, 1.1707, 0.9300])
  14. obser_var_biased: tensor([1.0613, 1.1239, 0.8928])
  15. obser_running_mean: tensor([ 0.0809,  0.1068, -0.0418])
  16. obser_running_var: tensor([1.0528, 1.0853, 0.9650])
  17. running_mean: tensor([ 0.0809,  0.1068, -0.0418])
  18. running_var: tensor([1.0528, 1.0853, 0.9650])
  19. output[0][0]: tensor([[-0.5016, -1.2827, -0.6372, -0.9518, -0.3242],
  20.         [-1.4549,  0.7529,  0.3830,  2.1300, -0.7114],
  21.         [ 0.2246,  2.4305,  0.1768, -1.0598,  0.0168],
  22.         [-1.1575,  0.0482,  0.3237,  1.5987, -0.1089],
  23.         [-1.1844,  0.2861, -0.2944,  1.0180,  0.2795]],
  24.        grad_fn=<SelectBackward0>)
  25. output_obser: tensor([[-0.5016, -1.2827, -0.6372, -0.9518, -0.3242],
  26.         [-1.4549,  0.7529,  0.3830,  2.1300, -0.7114],
  27.         [ 0.2246,  2.4305,  0.1768, -1.0598,  0.0168],
  28.         [-1.1575,  0.0482,  0.3237,  1.5987, -0.1089],
  29.         [-1.1844,  0.2861, -0.2944,  1.0180,  0.2795]])

参考：BatchNorm2d — PyTorch 2.0 documentation

详细解读nn.BatchNorm2d——批量标准化操作_ChaoFeiLi的博客-CSDN博客

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