batch normlization
see Sergey Ioffe and Christian Szegedy, "Batch Normalization: Accelerating Deep Network Training by Reducing
Internal Covariate Shift", ICML 2015.

首先完成前向過程：

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.                                          #
        #######################################################################
        sample_mean = np.mean(x,axis = 0) #矩陣x每一列的平均值(D,)
        sample_var = np.var(x,axis = 0) #矩陣x每一列的方差(D,)
        x_hat = (x - sample_mean)/(np.sqrt(sample_var + eps)) #標準化，eps:防止除數為0而增加的一個很小的正數
        out = gamma * x_hat + beta #gamma放縮係數，beta偏移常量
        cache = (x,sample_mean,sample_var,x_hat,eps,gamma,beta)
        running_mean = momentum * running_mean + (1 - momentum) * sample_mean #基於動量的引數衰減
        running_var = momentum * running_var + (1 - momentum) * sample_var
        #######################################################################
        #                           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.                               #
        #######################################################################
        out = (x - running_mean) * gamma / (np.sqrt(running_var + eps)) + 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
 

在訓練階段檢查：
Before batch normalization:
means: [ -2.3814598 -13.18038246 1.91780462]
stds: [27.18502186 34.21455511 37.68611762]

After batch normalization (gamma=1, beta=0)
means: [5.32907052e-17 7.04991621e-17 1.85962357e-17]
stds: [0.99999999 1. 1. ]

After batch normalization (gamma= [1. 2. 3.] , beta= [11. 12. 13.] )
means: [11. 12. 13.]
stds: [0.99999999 1.99999999 2.99999999]

發現means接近於beta（偏移），stds接近於gamma（標準化）.

在測試階段檢查：
After batch normalization (test-time):
means: [-0.03927354 -0.04349152 -0.10452688]
stds: [1.01531428 1.01238373 0.97819988]

仍然是means接近於1，stds接近於0，但比訓練階段的差別更大。

完成後向過程:

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.                                              #
    ###########################################################################
    #這部分就是利用公式計算梯度
    x,mean,var,x_hat,eps,gamma,beta = cache
    N = x.shape[0]
    dgamma = np.sum(dout * x_hat,axis = 0)
    dbeta = np.sum(dout * 1.0,axis = 0)
    dx_hat = dout * gamma
    dx_hat_numerator = dx_hat / np.sqrt(var + eps)
    dx_hat_denominator = np.sum(dx_hat * (x - mean),axis = 0)
    dx_1 = dx_hat_numerator
    dvar = -0.5 * ((var + eps) ** (-1.5)) * dx_hat_denominator
    dmean = -1.0 * np.sum(dx_hat_numerator,axis = 0) + dvar * np.mean(-2.0 * (x - mean),axis = 0)
    dx_var = dvar * 2.0 / N * (x - mean)
    dx_mean = dmean * 1.0 / N
    dx = dx_1 + dx_var + dx_mean
    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################

    return dx, dgamma, dbeta
 

dx error: 1.7029261167605239e-09
dgamma error: 7.420414216247087e-13
dbeta error: 2.8795057655839487e-12

使用論文中推導的公式來計算梯度:

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.#
    ###########################################################################
    #使用論文中推導的公式計算
    x,mean,var,x_hat,eps,gamma,beta = cache
    N = x.shape[0]
    dbeta = np.sum(dout,axis = 0)
    dgamma = np.sum(x_hat * dout,axis = 0)
    dx_norm = dout * gamma
    dv = ((x - mean) * -0.5 * (var + eps)**-1.5 * dx_norm).sum(axis = 0)
    dm = (dx_norm * -1 * (var + eps)** -0.5).sum(axis = 0) + (dv * (x - mean) * -2 / N).sum(axis = 0)
    dx = dx_norm / (var + eps) ** 0.5 + dv * 2 * (x - mean) / N + dm / N
    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################

    return dx, dgamma, dbeta

dx difference: 1.2266724949639026e-12
dgamma difference: 0.0
dbeta difference: 0.0
speedup: 0.83x

然而結果很尷尬，還不如不加速呢。。

測試有batchnorm的全連線網路：
多層的全連線網路：

class FullyConnectedNet(object):
    """
    A fully-connected neural network with an arbitrary number of hidden layers,
    ReLU nonlinearities, and a softmax loss function. This will also implement
    dropout and batch/layer normalization as options. For a network with L layers,
    the architecture will be

    {affine - [batch/layer norm] - relu - [dropout]} x (L - 1) - affine - softmax

    where batch/layer normalization and dropout are optional, and the {...} block is
    repeated L - 1 times.

    Similar to the TwoLayerNet above, learnable parameters are stored in the
    self.params dictionary and will be learned using the Solver class.
    """

    def __init__(self, hidden_dims, input_dim=3*32*32, num_classes=10,
                 dropout=1, normalization=None, reg=0.0,
                 weight_scale=1e-2, dtype=np.float32, seed=None):
        """
        Initialize a new FullyConnectedNet.

        Inputs:
        - hidden_dims: A list of integers giving the size of each hidden layer.
        - input_dim: An integer giving the size of the input.
        - num_classes: An integer giving the number of classes to classify.
        - dropout: Scalar between 0 and 1 giving dropout strength. If dropout=1 then
          the network should not use dropout at all.
        - normalization: What type of normalization the network should use. Valid values
          are "batchnorm", "layernorm", or None for no normalization (the default).
        - reg: Scalar giving L2 regularization strength.
        - weight_scale: Scalar giving the standard deviation for random
          initialization of the weights.
        - dtype: A numpy datatype object; all computations will be performed using
          this datatype. float32 is faster but less accurate, so you should use
          float64 for numeric gradient checking.
        - seed: If not None, then pass this random seed to the dropout layers. This
          will make the dropout layers deteriminstic so we can gradient check the
          model. 預設無隨機種子，若有會傳遞給dropout層。
        """
        self.normalization = normalization
        self.use_dropout = dropout != 1
        self.reg = reg
        self.num_layers = 1 + len(hidden_dims)
        self.dtype = dtype
        self.params = {}

        ############################################################################
        # TODO: Initialize the parameters of the network, storing all values in    #
        # the self.params dictionary. Store weights and biases for the first layer #
        # in W1 and b1; for the second layer use W2 and b2, etc. Weights should be #
        # initialized from a normal distribution centered at 0 with standard       #
        # deviation equal to weight_scale. Biases should be initialized to zero.   #
        #                                                                          #
        # When using batch normalization, store scale and shift parameters for the #
        # first layer in gamma1 and beta1; for the second layer use gamma2 and     #
        # beta2, etc. Scale parameters should be initialized to ones and shift     #
        # parameters should be initialized to zeros.                               #
        ############################################################################
        #初始化所有隱藏層的引數
        in_dim = input_dim #D
        for i,h_dim in enumerate(hidden_dims): #(0,H1)(1,H2)
            self.params['W%d' %(i+1,)] = weight_scale * np.random.randn(in_dim,h_dim)
            self.params['b%d' %(i+1,)] = np.zeros((h_dim,))
            if self.normalization=='batchnorm':
                self.params['gamma%d' %(i+1,)] = np.ones((h_dim,)) #初始化為1
                self.params['beta%d' %(i+1,)] = np.zeros((h_dim,)) #初始化為0
            in_dim = h_dim #將該層的列數傳遞給下一層的行數
            
        #初始化所有輸出層的引數
        self.params['W%d' %(self.num_layers,)] = weight_scale * np.random.randn(in_dim,num_classes)
        self.params['b%d' %(self.num_layers,)] = np.zeros((num_classes,))
        ############################################################################
        #                             END OF YOUR CODE                             #
        ############################################################################

        #  當開啟 dropout 時，我們需要在每一個神經元層中傳遞一個相同的 dropout 引數字典 self.dropout_param ，以保證每一層的神經元們 都知曉失活概率p和當前神經網路的模式狀態mode(訓練／測試)。 
        self.dropout_param = {} #dropout的引數字典
        if self.use_dropout:
            self.dropout_param = {'mode': 'train', 'p': dropout}
            if seed is not None:
                self.dropout_param['seed'] = seed

        #  當開啟批量歸一化時，我們要定義一個BN演算法的引數列表 self.bn_params ， 以用來跟蹤記錄每一層的平均值和標準差。其中，第0個元素 self.bn_params[0] 表示前向傳播第1個BN層的引數，第1個元素 self.bn_params[1] 表示前向傳播 第2個BN層的引數，以此類推。
        self.bn_params = [] #BN的引數字典
        if self.normalization=='batchnorm':
            self.bn_params = [{'mode': 'train'} for i in range(self.num_layers - 1)]
        if self.normalization=='layernorm':
            self.bn_params = [{} for i in range(self.num_layers - 1)]

        # Cast all parameters to the correct datatype
        for k, v in self.params.items():
            self.params[k] = v.astype(dtype)


    def loss(self, X, y=None):
        """
        Compute loss and gradient for the fully-connected net.

        Input / output: Same as TwoLayerNet above.
        """
        X = X.astype(self.dtype)
        mode = 'test' if y is None else 'train'

        # Set train/test mode for batchnorm params and dropout param since they
        # behave differently during training and testing.
        if self.use_dropout:
            self.dropout_param['mode'] = mode
        if self.normalization=='batchnorm':
            for bn_param in self.bn_params:
                bn_param['mode'] = mode
        scores = None
        ############################################################################
        # TODO: Implement the forward pass for the fully-connected net, computing  #
        # the class scores for X and storing them in the scores variable.          #
        #                                                                          #
        # When using dropout, you'll need to pass self.dropout_param to each       #
        # dropout forward pass.                                                    #
        #                                                                          #
        # When using batch normalization, you'll need to pass self.bn_params[0] to #
        # the forward pass for the first batch normalization layer, pass           #
        # self.bn_params[1] to the forward pass for the second batch normalization #
        # layer, etc.                                                              #
        ############################################################################
        fc_mix_cache = {} # # 初始化每層前向傳播的緩衝字典
        if self.use_dropout: # 如果開啟了dropout，初始化其對應的緩衝字典
            dp_cache = {}
        # 從第一個隱藏層開始迴圈每一個隱藏層，傳遞資料out，儲存每一層的緩衝cache
        out = X
        for i in range(self.num_layers - 1): # 在每個hidden層中迴圈
            w,b = self.params['W%d' %(i+1,)],self.params['b%d' %(i+1,)]
            if self.normalization == 'batchnorm':
                gamma = self.params['gamma%d' %(i+1,)]
                beta = self.params['beta%d' %(i+1,)]
                out,fc_mix_cache[i] = affine_bn_relu_forward(out,w,b,gamma,beta,self.bn_params[i])
            else:
                out,fc_mix_cache[i] = affine_relu_forward(out,w,b)
            if self.use_dropout:
                out,dp_cache[i] = dropout_forward(out,self.dropout_param)
        #最後的輸出層
        w = self.params['W%d' %(self.num_layers,)]
        b = self.params['b%d' %(self.num_layers,)]
        out,out_cache = affine_forward(out,w,b)
        scores = out
        ############################################################################
        #                             END OF YOUR CODE                             #
        ############################################################################

        # If test mode return early
        if mode == 'test':
            return scores

        loss, grads = 0.0, {}
        ############################################################################
        # TODO: Implement the backward pass for the fully-connected net. Store the #
        # loss in the loss variable and gradients in the grads dictionary. Compute #
        # data loss using softmax, and make sure that grads[k] holds the gradients #
        # for self.params[k]. Don't forget to add L2 regularization!               #
        #                                                                          #
        # When using batch/layer normalization, you don't need to regularize the scale   #
        # and shift parameters.                                                    #
        #                                                                          #
        # NOTE: To ensure that your implementation matches ours and you pass the   #
        # automated tests, make sure that your L2 regularization includes a factor #
        # of 0.5 to simplify the expression for the gradient.                      #
        ############################################################################
        loss,dout = softmax_loss(scores,y)
        loss += 0.5 * self.reg * np.sum(self.params['W%d' %(self.num_layers,)] ** 2)
        # 在輸出層處梯度的反向傳播，順便把梯度儲存在梯度字典 grad 中：
        dout,dw,db = affine_backward(dout,out_cache)
        grads['W%d' %(self.num_layers,)] = dw + self.reg * self.params['W%d' %(self.num_layers,)]
        grads['b%d' %(self.num_layers,)] = db
        # 在每一個隱藏層處梯度的反向傳播，不僅順便更新了梯度字典 grad，還迭代算出了損失值loss
        for i in range(self.num_layers - 1):
            ri = self.num_layers - 2 - i #倒數第ri+1隱藏層
            loss += 0.5 * self.reg * np.sum(self.params['W%d' %(ri+1,)] ** 2) #迭代地補上每層的正則項給loss
            if self.use_dropout:
                dout = dropout_backward(dout,dp_cache[ri])
            if self.normalization == 'batchnorm':
                dout,dw,db,dgamma,dbeta = affine_bn_relu_backward(dout,fc_mix_cache[ri])
                grads['gamma%d' %(ri+1,)] = dgamma
                grads['beta%d' %(ri+1,)] = dbeta
            else:
                dout,dw,db = affine_relu_backward(dout,fc_mix_cache[ri])
            grads['W%d' %(ri+1,)] = dw + self.reg * self.params['W%d' %(ri+1,)]
            grads['b%d' %(ri+1,)] = db
        ############################################################################
        #                             END OF YOUR CODE                             #
        ############################################################################

        return loss, grads

用到的兩個結構：

def affine_bn_relu_forward(x, w, b, gamma, beta, bn_param):
    affine_out, fc_cache = affine_forward(x, w, b) #線性模型
    bn_out, bn_cache = batchnorm_forward(affine_out, gamma, beta, bn_param) #batchnorm
    relu_out, relu_cache = relu_forward(bn_out) #relu層
    cache = (fc_cache, bn_cache, relu_cache)  
    return relu_out, cache


def affine_bn_relu_backward(dout, cache):
    fc_cache, bn_cache, relu_cache = cache 
    drelu_out = relu_backward(dout, relu_cache) #relu
    dbn_out, dgamma, dbeta = batchnorm_backward(drelu_out, bn_cache) #batchnorm
    dx, dw, db = affine_backward(dbn_out, fc_cache)  #線性
    return dx, dw, db, dgamma, dbeta

Running check with reg = 0
Initial loss: 2.2611955101340957
W1 relative error: 1.10e-04
W2 relative error: 2.85e-06
W3 relative error: 4.05e-10
b1 relative error: 4.44e-08
b2 relative error: 2.22e-08
b3 relative error: 1.01e-10
beta1 relative error: 7.33e-09
beta2 relative error: 1.89e-09
gamma1 relative error: 6.96e-09
gamma2 relative error: 1.96e-09

Running check with reg = 3.14
Initial loss: 6.996533220108303
W1 relative error: 1.98e-06
W2 relative error: 2.28e-06
W3 relative error: 1.11e-08
b1 relative error: 2.78e-09
b2 relative error: 2.22e-08
b3 relative error: 2.10e-10
beta1 relative error: 6.65e-09
beta2 relative error: 4.23e-09
gamma1 relative error: 6.27e-09
gamma2 relative error: 5.28e-09

用一個六層的網路，10000個訓練樣本來檢測batchnorm的效果

發現當然是有batchnorm的效果更好

初始化和batchnorm的互動作用

發現有batchnorm的情況下，權重初始化用訓練的影響更小，或者說，batchnorm幫助控制了當權重初始化不好的情況，專業點的話，有batchnorm對權重初始化的魯棒性更高。。
在權重初始化不好的情況下，batchnorm的效能明顯優於沒有baseline；權重初始化較好的情況下，表現相當。

研究batchnorm和batch size之間的關係：

發現batch越大結果越好（當然了）。batch越大對於平均值和方差的估計越準確，結果越好。所以在硬體支援的情況下，batch越大越好。

Layer Normalization:
LN是針對深度網路的某一層的所有神經元的輸入進行normalize操作。LN中同層神經元輸入擁有相同的均值和方差，不同的輸入樣本有不同的均值和方差。LN用於RNN效果比較明顯，但是在CNN上，不如BN。

see Ba, Jimmy Lei, Jamie Ryan Kiros, and Geoffrey E. Hinton. "Layer Normalization." stat 1050 (2016): 21.

前向傳播：

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?                      #
    ###########################################################################
    #layer norm是對輸入的資料的每一個樣本（1，D），求均值方差等等，不依賴batch。相比batch norm有自己的特點。但彷彿在卷積網路中效果不如batch norm
    x_T = x.T
    sample_mean = np.mean(x_T,axis = 0)
    sample_var = np.var(x_T,axis = 0)
    x_norm_T = (x_T - sample_mean) / np.sqrt(sample_var + eps)
    x_norm = x_norm_T.T
    out = x_norm * gamma + beta
    cache = (x,x_norm,gamma,sample_mean,sample_var,eps)
    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################
    return out, cache

Before layer normalization:
means: [-59.06673243 -47.60782686 -43.31137368 -26.40991744]
stds: [10.07429373 28.39478981 35.28360729 4.01831507]

After layer normalization (gamma=1, beta=0)
means: [ 4.81096644e-16 -7.40148683e-17 2.22044605e-16 -5.92118946e-16]
stds: [0.99999995 0.99999999 1. 0.99999969]

After layer normalization (gamma= [3. 3. 3.] , beta= [5. 5. 5.] )
means: [5. 5. 5. 5.]
stds: [2.99999985 2.99999998 2.99999999 2.99999907]

後向傳播：

def 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_norm,gamma,sample_mean,sample_var,eps = cache
    x_T = x.T
    dout_T = dout.T
    N = x_T.shape[0]
    dbeta = np.sum(dout,axis = 0)
    dgamma = np.sum(x_norm * dout,axis = 0)
    dx_norm = dout_T * gamma[:,np.newaxis]
    dv = ((x_T - sample_mean) * -0.5 * (sample_var + eps)** -1.5 * dx_norm).sum(axis = 0)
    dm = (dx_norm * -1 * (sample_var + eps) ** -0.5).sum(axis = 0) + (dv * (x_T - sample_mean) * -2 / N).sum(axis = 0)
    dx_T = dx_norm / (sample_var + eps)** 0.5 + dv * 2 * (x_T - sample_mean) / N + dm / N
    dx = dx_T.T
    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################
    return dx, dgamma, dbeta

dx error: 1.433615657860454e-09
dgamma error: 4.519489546032799e-12
dbeta error: 2.276445013433725e-12

layernorm和batch size之間的關係：

發現與batch的大小沒有關係。