Question

根据输入值[1, 5]并对其进行规范化，应该产生类似[-1, 1] if I understand correctly的内容，因为

mean = 3
var = 4
result = (x - mean) / sqrt(var)

然而这个最小的例子

import numpy as np

import keras
from keras.models import Model
from keras.layers import Input
from keras.layers.normalization import BatchNormalization
from keras import backend as K

shape = (1,2,1)
input = Input(shape=shape)
x = BatchNormalization(center=False)(input) # no beta
model = Model(inputs=input, outputs=x)
model.compile(loss='mse', optimizer='sgd')

# training with dummy data
training_in = [np.random.random(size=(10, *shape))]
training_out = [np.random.random(size=(10, *shape))]
model.fit(training_in, training_out, epochs=10)

data_in = np.array([[[[1], [5]]]], dtype=np.float32)
data_out = model.predict(data_in)

print('gamma   :', K.eval(model.layers[1].gamma))
#print('beta    :', K.eval(model.layers[1].beta))
print('moving_mean:', K.eval(model.layers[1].moving_mean))
print('moving_variance:', K.eval(model.layers[1].moving_variance))

print('epsilon :', model.layers[1].epsilon)
print('data_in :', data_in)
print('data_out:', data_out)

产生以下输出：

gamma   : [ 0.80644524]
moving_mean: [ 0.05885344]
moving_variance: [ 0.91000736]
epsilon : 0.001
data_in : [[[[ 1.]
   [ 5.]]]]
data_out: [[[[ 0.79519051]
   [ 4.17485714]]]]

所以它是[0.79519051, 4.17485714]而不是[-1, 1]。

我查看了source，这些值似乎转发给了tf.nn.batch_normalization。而looks这样的结果应该是我的意思，但显然不是。

那么如何计算输出值？

Answer 1

如果您正在使用gamma，则正确的等式实际上是result = gamma * (x - mean) / sqrt(var)用于批量规范化， BUT mean和var是并不总是一样：

在培训（适合）期间，使用批次的输入值计算mean_batch和var_batch（它们只是批次的均值和方差）），就像您和＃39;正在做。同时，全球moving_mean和moving_variance以这种方式学习：moving_mean = alpha * moving_mean + (1-alpha) * mean_batch，alpha是一种学习率，在（0,1）中，通常高于0.9。 moving_mean和moving_variance是所有培训数据的实际均值和方差的近似值。通过通常的梯度下降，我们也可以学习Gamma，以最适合您的输出。
在推理（预测）期间，您只需使用moving_mean和moving_variance的学习值，而不是mean_batch和var_batch。您还可以使用学习过的gamma。

所以0.05885344只是随机输入数据的平均值的近似值，它的方差为0.91000736，您可以使用它们来标准化新数据[1,5]。您可以轻松查看[0.79519051, 4.17485714]=gamma * ([1, 5] - moving_mean)/sqrt(moving_var)

编辑：alpha在keras中被称为动量，如果你想检查它。

Answer 2

正确的公式是：

result = gamma * (input - moving_mean) / sqrt(moving_variance + epsilon) + beta

这是一个验证脚本：

import math
import numpy as np
import tensorflow as tf
from keras import backend as K

from keras.models import Model
from keras.layers import Input
from keras.layers.normalization import BatchNormalization

np.random.seed(0)

print('=== keras model ===')
input_shape = (1,2,1)
input = Input(shape=input_shape)
x = BatchNormalization()(input)
model = Model(inputs=input, outputs=x)
model.compile(loss='mse', optimizer='sgd')
training_in = [np.random.random(size=(10, *input_shape))]
training_out = [np.random.random(size=(10, *input_shape))]
model.fit(training_in, training_out, epochs=100, verbose=0)
data_in = [[[1.0], [5.0]]]
data_model = np.array([data_in])
result = model.predict(data_model)
gamma = K.eval(model.layers[1].gamma)
beta = K.eval(model.layers[1].beta)
moving_mean = K.eval(model.layers[1].moving_mean)
moving_variance = K.eval(model.layers[1].moving_variance)
epsilon = model.layers[1].epsilon
print('gamma:          ', gamma)
print('beta:           ', beta)
print('moving_mean:    ', moving_mean)
print('moving_variance:', moving_variance)
print('epsilon:        ', epsilon)
print('data_in:        ', data_in)
print('result:         ', result)

print('=== numpy ===')
np_data = [data_in[0][0][0], data_in[0][1][0]]
np_mean = moving_mean[0]
np_variance = moving_variance[0]
np_offset = beta[0]
np_scale = gamma[0]
np_result = [np_scale * (x - np_mean) / math.sqrt(np_variance + epsilon) + np_offset for x in np_data]
print(np_result)

print('=== tensorflow ===')
tf_data = tf.constant(data_in)
tf_mean = tf.constant(moving_mean)
tf_variance = tf.constant(moving_variance)
tf_offset = tf.constant(beta)
tf_scale = tf.constant(gamma)
tf_variance_epsilon = epsilon
tf_result = tf.nn.batch_normalization(tf_data, tf_mean, tf_variance, tf_offset, tf_scale, tf_variance_epsilon)
tf_sess = tf.Session()
print(tf_sess.run(tf_result))

print('=== keras backend ===')
k_data = K.constant(data_in)
k_mean = K.constant(moving_mean)
k_variance = K.constant(moving_variance)
k_offset = K.constant(beta)
k_scale = K.constant(gamma)
k_variance_epsilon = epsilon
k_result = K.batch_normalization(k_data, k_mean, k_variance, k_offset, k_scale, k_variance_epsilon)
print(K.eval(k_result))

输出：

gamma:           [ 0.22297101]
beta:            [ 0.49253803]
moving_mean:     [ 0.36868709]
moving_variance: [ 0.41429576]
epsilon:         0.001
data_in:         [[[1.0], [5.0]]]
result:          [[[[ 0.71096909]
   [ 2.09494853]]]]

=== numpy ===
[0.71096905498374263, 2.0949484904433255]

=== tensorflow ===
[[[ 0.71096909]
  [ 2.09494853]]]

=== keras backend ===
[[[ 0.71096909]
  [ 2.09494853]]]

Keras中的BatchNormalization图层提供了意外的输出值

2 个答案: