# Batch Normalization&Dropout浅析

## 一. Batch Normalization

#### 代码实现：

```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.                                                          #
#######################################################################
sample_mean = x.mean(axis = 0)
sample_var = x.var(axis = 0)
x_hat = (x-sample_mean)/(np.sqrt(sample_var+eps))
out = gamma*x_hat+beta
running_mean = momentum * running_mean + (1 - momentum) * sample_mean
running_var = momentum * running_var + (1 - momentum) * sample_var

#cache = (x,gamma,beta)
cache = (gamma, x, sample_mean, sample_var, eps, x_hat)
#######################################################################
#                           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_h = (x-bn_param["running_mean"])/(np.sqrt(bn_param["running_var"]+eps))
out = gamma*x_h+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.                         #
###########################################################################
gamma, x, sample_mean, sample_var, eps, x_hat = cache
N = x.shape[0]
D = x.shape[1]
dgamma = np.sum(dout * x_hat,axis = 0)#(D,)

dbeta = dout.sum(axis = 0)#(D,)
dx_hat = dout * gamma#(N,D)
std = np.sqrt(sample_var.reshape(1,D) + eps)#(1,D)
dx = dx_hat / std#(N,D)
dstd = np.sum(-dx_hat*(x_hat/std),axis = 0).reshape(1,D)#(1,D)
dm = np.sum(-dx_hat / std,axis = 0).reshape(1,D)#(1,D)
dvar = dstd/(2*std)#(1,D)
dm += dvar*(-2/N)*((x-sample_mean).sum(axis = 0).reshape(1,D))#(1,D)
dx += dvar * (2/N)*(x-sample_mean)#(N,D)
dx += dm / N#(N,D)
###########################################################################
#                             END OF YOUR CODE                            #
###########################################################################

return dx, dgamma, dbeta```

## 二. Dropout

Dropout 背后的思想其实就是把DNN当做一个集成模型来训练，之后取所有值的平均值，而不只是训练单个DNN。

DNN网络将Dropout率设置为 p，也就是说，一个神经元被保留的概率是 1-p。当一个神经元被丢弃时，无论输入或者相关的参数是什么，它的输出值就会被设置为0。

### - Inverted Dropout（Dropout 改进版）

#### 代码实现：

```def dropout_forward(x, dropout_param):
"""
Performs the forward pass for (inverted) dropout.

Inputs:
- x: Input data, of any shape
- dropout_param: A dictionary with the following keys:
- p: Dropout parameter. We drop each neuron output with probability p.
- mode: 'test' or 'train'. If the mode is train, then perform dropout;
if the mode is test, then just return the input.
- seed: Seed for the random number generator. Passing seed makes this
function deterministic, which is needed for gradient checking but not
in real networks.

Outputs:
- out: Array of the same shape as x.
- cache: tuple (dropout_param, mask). In training mode, mask is the dropout
mask that was used to multiply the input; in test mode, mask is None.
"""
p, mode = dropout_param['p'], dropout_param['mode']
if 'seed' in dropout_param:
np.random.seed(dropout_param['seed'])

out = None

if mode == 'train':
#######################################################################
# TODO: Implement training phase forward pass for inverted dropout.   #
#######################################################################
#musk = np.random.rand(*x.shape) >= p
mask = (np.random.rand(*x.shape) >= p) / (1 - p)
#######################################################################
#                           END OF YOUR CODE                          #
#######################################################################
elif mode == 'test':
#######################################################################
# TODO: Implement the test phase forward pass for inverted dropout.   #
#######################################################################
out = x
#######################################################################
#                            END OF YOUR CODE                         #
#######################################################################

out = out.astype(x.dtype, copy=False)

return out, cache

def dropout_backward(dout, cache):
"""
Perform the backward pass for (inverted) dropout.

Inputs:
- dout: Upstream derivatives, of any shape
- cache: (dropout_param, mask) from dropout_forward.
"""
mode = dropout_param['mode']

dx = None
if mode == 'train':
#######################################################################
# TODO: Implement training phase backward pass for inverted dropout   #
#######################################################################
#######################################################################
#                          END OF YOUR CODE                           #
#######################################################################
elif mode == 'test':
dx = dout
return dx```

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