版权声明：王家林大咖2018年新书《SPARK大数据商业实战三部曲》清华大学出版，清华大学出版社官方旗舰店（天猫）https://qhdx.tmall.com/?spm=a220o.1000855.1997427721.d4918089.4b2a2e5dT6bUsM https://blog.csdn.net/duan_zhihua/article/details/83107615
cs 231 Batch Normalization 求导推导及代码复现：
作者论文公式：https://arxiv.org/abs/1502.03167
Batch Normalization 计算图：
Batch Normalization 求导数学推导：
Batch Normalization 对xi 三条路径最终推出的结果：
论文公式代码复现如下：
def batchnorm_forward(x, gamma, beta, bn_param):
"""
Forward pass for batch normalization.
During training the sample mean and (uncorrected) sample variance are
computed from minibatch statistics and used to normalize the incoming data.
During training we also keep an exponentially decaying running mean of the
mean and variance of each feature, and these averages are used to normalize
data at test-time.
At each timestep we update the running averages for mean and variance using
an exponential decay based on the momentum parameter:
running_mean = momentum * running_mean + (1 - momentum) * sample_mean
running_var = momentum * running_var + (1 - momentum) * sample_var
Note that the batch normalization paper suggests a different test-time
behavior: they compute sample mean and variance for each feature using a
large number of training images rather than using a running average. For
this implementation we have chosen to use running averages instead since
they do not require an additional estimation step; the torch7
implementation of batch normalization also uses running averages.
Input:
- x: Data of shape (N, D)
- gamma: Scale parameter of shape (D,)
- beta: Shift paremeter of shape (D,)
- bn_param: Dictionary with the following keys:
- mode: 'train' or 'test'; required
- eps: Constant for numeric stability
- momentum: Constant for running mean / variance.
- running_mean: Array of shape (D,) giving running mean of features
- running_var Array of shape (D,) giving running variance of features
Returns a tuple of:
- out: of shape (N, D)
- cache: A tuple of values needed in the backward pass
"""
mode = bn_param['mode']
eps = bn_param.get('eps', 1e-5)
momentum = bn_param.get('momentum', 0.9)
N, D = x.shape
running_mean = bn_param.get('running_mean', np.zeros(D, dtype=x.dtype))
running_var = bn_param.get('running_var', np.zeros(D, dtype=x.dtype))
out, cache = None, None
if mode == 'train':
#######################################################################
# TODO: Implement the training-time forward pass for batch norm. #
# Use minibatch statistics to compute the mean and variance, use #
# these statistics to normalize the incoming data, and scale and #
# shift the normalized data using gamma and beta. #
# #
# You should store the output in the variable out. Any intermediates #
# that you need for the backward pass should be stored in the cache #
# variable. #
# #
# You should also use your computed sample mean and variance together #
# with the momentum variable to update the running mean and running #
# variance, storing your result in the running_mean and running_var #
# variables. #
# #
# Note that though you should be keeping track of the running #
# variance, you should normalize the data based on the standard #
# deviation (square root of variance) instead! #
# Referencing the original paper (https://arxiv.org/abs/1502.03167) #
# might prove to be helpful. #
#######################################################################
#公式： https://arxiv.org/abs/1502.03167
mean_x = np.mean(x, axis = 0 )
var_x = np.var(x, axis = 0)
x_hat =( x - mean_x) / np.sqrt(var_x + eps )
out = gamma* x_hat + beta
running_mean = momentum * running_mean + (1 - momentum) * mean_x
running_var = momentum * running_var + (1 - momentum) * var_x
inv_var_x = 1 / np.sqrt(var_x + eps)
cache =(x,x_hat,gamma,mean_x,inv_var_x)
#######################################################################
# END OF YOUR CODE #
#######################################################################
elif mode == 'test':
#######################################################################
# TODO: Implement the test-time forward pass for batch normalization. #
# Use the running mean and variance to normalize the incoming data, #
# then scale and shift the normalized data using gamma and beta. #
# Store the result in the out variable. #
#######################################################################
x_hat =( x - running_mean) / np.sqrt(running_var + eps )
out = gamma* x_hat + beta
#######################################################################
# END OF YOUR CODE #
#######################################################################
else:
raise ValueError('Invalid forward batchnorm mode "%s"' % mode)
# Store the updated running means back into bn_param
bn_param['running_mean'] = running_mean
bn_param['running_var'] = running_var
return out, cache
def batchnorm_backward(dout, cache):
"""
Backward pass for batch normalization.
For this implementation, you should write out a computation graph for
batch normalization on paper and propagate gradients backward through
intermediate nodes.
Inputs:
- dout: Upstream derivatives, of shape (N, D)
- cache: Variable of intermediates from batchnorm_forward.
Returns a tuple of:
- dx: Gradient with respect to inputs x, of shape (N, D)
- dgamma: Gradient with respect to scale parameter gamma, of shape (D,)
- dbeta: Gradient with respect to shift parameter beta, of shape (D,)
"""
dx, dgamma, dbeta = None, None, None
###########################################################################
# TODO: Implement the backward pass for batch normalization. Store the #
# results in the dx, dgamma, and dbeta variables. #
# Referencing the original paper (https://arxiv.org/abs/1502.03167) #
# might prove to be helpful. #
###########################################################################
# =============================================================================
# xi ----- uB----- o^2 B------------xi^--------------yi----------l
# xi-----
# ub---- gamma--
# xi---- betla--
# =============================================================================
x, x_hat, gamma, mu, inv_sigma = cache
x,x_hat,gamma,mean_x,inv_var_x = cache
N = x.shape[0]
# dx 求导合并：
#1: l--->xi^--->xi
dx= gamma * dout * inv_var_x
#2: l----> o^2 B--->xi
dx += (2 / N) * (x - mean_x) * np.sum(- (1/2) * inv_var_x ** 3 * (x - mean_x) * gamma * dout, axis=0)
#3: l----> uB--->xi
dx += (1 / N) * np.sum(-1 * inv_var_x * gamma * dout, axis=0)
# dgamma求导：l----> yi--->gamma
dgamma = np.sum(x_hat * dout, axis=0)
# dbeta求导：l----> yi--->betla
dbeta = np.sum(dout, axis=0)
###########################################################################
# END OF YOUR CODE #
###########################################################################
return dx, dgamma, dbeta
batchnorm_backward_alt
代码复现如下：
def batchnorm_backward_alt(dout, cache):
"""
Alternative backward pass for batch normalization.
For this implementation you should work out the derivatives for the batch
normalizaton backward pass on paper and simplify as much as possible. You
should be able to derive a simple expression for the backward pass.
See the jupyter notebook for more hints.
Note: This implementation should expect to receive the same cache variable
as batchnorm_backward, but might not use all of the values in the cache.
Inputs / outputs: Same as batchnorm_backward
"""
dx, dgamma, dbeta = None, None, None
###########################################################################
# TODO: Implement the backward pass for batch normalization. Store the #
# results in the dx, dgamma, and dbeta variables. #
# #
# After computing the gradient with respect to the centered inputs, you #
# should be able to compute gradients with respect to the inputs in a #
# single statement; our implementation fits on a single 80-character line.#
###########################################################################
x, x_hat, gamma, mean_x,inv_var_x = cache
N = x.shape[0]
dbeta = np.sum(dout, axis=0)
dgamma = np.sum(x_hat * dout, axis=0)
dxhat = dout * gamma
dx = (1. / N) * inv_var_x * (N * dxhat - np.sum(dxhat, axis=0) -
x_hat * np.sum(dxhat * x_hat, axis=0))
###########################################################################
# END OF YOUR CODE #
###########################################################################
return dx, dgamma, dbeta
layer normalization:
def layernorm_forward(x, gamma, beta, ln_param):
"""
Forward pass for layer normalization.
During both training and test-time, the incoming data is normalized per data-point,
before being scaled by gamma and beta parameters identical to that of batch normalization.
Note that in contrast to batch normalization, the behavior during train and test-time for
layer normalization are identical, and we do not need to keep track of running averages
of any sort.
Input:
- x: Data of shape (N, D)
- gamma: Scale parameter of shape (D,)
- beta: Shift paremeter of shape (D,)
- ln_param: Dictionary with the following keys:
- eps: Constant for numeric stability
Returns a tuple of:
- out: of shape (N, D)
- cache: A tuple of values needed in the backward pass
"""
out, cache = None, None
eps = ln_param.get('eps', 1e-5)
###########################################################################
# TODO: Implement the training-time forward pass for layer norm. #
# Normalize the incoming data, and scale and shift the normalized data #
# using gamma and beta. #
# HINT: this can be done by slightly modifying your training-time #
# implementation of batch normalization, and inserting a line or two of #
# well-placed code. In particular, can you think of any matrix #
# transformations you could perform, that would enable you to copy over #
# the batch norm code and leave it almost unchanged? #
###########################################################################
#x: (N, D) ---->(D,N)
x = x.T
mean_x = np.mean(x,axis =0)
var_x= np.var(x,axis = 0)
inv_var_x = 1 / np.sqrt(var_x + eps)
x_hat = (x - mean_x)/np.sqrt(var_x + eps) #(D,N)
x_hat = x_hat.T #(D,N)---->(N,D)
# gamma: (D,) beta: (D,)
out = gamma * x_hat + beta
cache =(x_hat,gamma,mean_x,inv_var_x)
###########################################################################
# END OF YOUR CODE #
###########################################################################
return out, cachedef layernorm_backward(dout, cache):
"""
Backward pass for layer normalization.
For this implementation, you can heavily rely on the work you've done already
for batch normalization.
Inputs:
- dout: Upstream derivatives, of shape (N, D)
- cache: Variable of intermediates from layernorm_forward.
Returns a tuple of:
- dx: Gradient with respect to inputs x, of shape (N, D)
- dgamma: Gradient with respect to scale parameter gamma, of shape (D,)
- dbeta: Gradient with respect to shift parameter beta, of shape (D,)
"""
dx, dgamma, dbeta = None, None, None
###########################################################################
# TODO: Implement the backward pass for layer norm. #
# #
# HINT: this can be done by slightly modifying your training-time #
# implementation of batch normalization. The hints to the forward pass #
# still apply! #
###########################################################################
x, x_hat, gamma, mean_x,inv_var_x = cache
d = x.shape[0]
dbeta = np.sum(dout, axis=0)
dgamma = np.sum(x_hat * dout, axis=0)
dxhat = dout * gamma
dxhat = dxhat.T
x_hat = x_hat.T
dx = (1. / d) * inv_var_x * (d * dxhat - np.sum(dxhat, axis=0) -
x_hat * np.sum(dxhat * x_hat, axis=0))
dx = dx.T
###########################################################################
# END OF YOUR CODE #
###########################################################################
return dx, dgamma, dbeta