Implementing Batch normalisation in Neural network

Question

I have implemented my own mini neural network program1. Currently, it does not have batch updates, it only updates the parameters by simple backpropagation using SGD after each forward pass. I was trying to implement batch updates and batch normalisation2.

1) For simple batch updates, instead of updating parameters each time, for each image of the batch size of 'n' I should backpropagate and accumulate the deltas for all the parameters and finally update them once after the end of the batch.

2)For batch normalisation (BN), I went through the paper and I am sort of the clear with the idea but I am confused regarding how to implement it. Generally, I would multiply the matrices in the net one after the other for a single image to get final input, but with BN, do I need to feed forward for all the images in the batch till the first layer, then normalise the values, then fwd pass these values till second layer, then normalise again, and so on? Once I reach the final layer, should I backpropagate the error for the corresponding input-output pair and update the parameters immediately as fwd pass for all the images in the batch has been done already?

Going by the way I have described, it seems to require a lot of parameter tracking throughout the batch. It will be helpful if you can point out a better way to do it or anything that I have misunderstood so far.

Answer 1

Give the whole batch as an input, take the activated output of the first layer and pipe it the next until the final. .

calculate the gradient decent direction (derivative of the cost function) for each row. U will have the error for each row of ur training set.

Then add the errors and divide the result with number of the training set. Then finally update it weights. I know from experience that tracking all these numbers is hard.

I don't know what language u use, but if it's Python, numpy will help u a a lot. For example, to add the errors, errors.sum(axis=0). X.dot(weight) to multiply ur entire dataset with ur weight. So my suggestion is use some matrix library.

One thing I want to mention is batch processing is susceptible to local minima which u don't want...especially in complex functions. SDG is a nice algorithm. I'm not sure why u needed batch update. There are many interesting tweaks in NN, don't waste ur time on batch update, use SDG

Answer 2

Explanation using the article Understanding the backward pass through Batch Normalization Layer and from the cs231n materials.

• cs231n 2020 lecture 7 slide pdf
• cs231n 2020 assignment 2 Batch Normalization

Batch data

D features x N number of data in batch is normalized is explained in the cs231n lecture 7 slide.

Forward and backward propagations

The forward and back propagation is in the diagram from Understanding the backward pass through Batch Normalization Layer.

Gradient from the softmax log loss layer

• Computing the Analytic Gradient with Backpropagation

Code

From Understanding the backward pass through Batch Normalization Layer. The step number matches with the number in the forward/backward diagram above.

Forward

def batchnorm_forward(x, gamma, beta, eps):

  N, D = x.shape

  #step1: calculate mean
  mu = 1./N * np.sum(x, axis = 0)

  #step2: subtract mean vector of every trainings example
  xmu = x - mu

  #step3: following the lower branch - calculation denominator
  sq = xmu ** 2

  #step4: calculate variance
  var = 1./N * np.sum(sq, axis = 0)

  #step5: add eps for numerical stability, then sqrt
  sqrtvar = np.sqrt(var + eps)

  #step6: invert sqrtwar
  ivar = 1./sqrtvar

  #step7: execute normalization
  xhat = xmu * ivar

  #step8: Nor the two transformation steps
  gammax = gamma * xhat

  #step9
  out = gammax + beta

  #store intermediate
  cache = (xhat,gamma,xmu,ivar,sqrtvar,var,eps)

  return out, cache

Backward

def batchnorm_backward(dout, cache):

  #unfold the variables stored in cache
  xhat,gamma,xmu,ivar,sqrtvar,var,eps = cache

  #get the dimensions of the input/output
  N,D = dout.shape

  #step9
  dbeta = np.sum(dout, axis=0)
  dgammax = dout #not necessary, but more understandable

  #step8
  dgamma = np.sum(dgammax*xhat, axis=0)
  dxhat = dgammax * gamma

  #step7
  divar = np.sum(dxhat*xmu, axis=0)
  dxmu1 = dxhat * ivar

  #step6
  dsqrtvar = -1. /(sqrtvar**2) * divar

  #step5
  dvar = 0.5 * 1. /np.sqrt(var+eps) * dsqrtvar

  #step4
  dsq = 1. /N * np.ones((N,D)) * dvar

  #step3
  dxmu2 = 2 * xmu * dsq

  #step2
  dx1 = (dxmu1 + dxmu2)
  dmu = -1 * np.sum(dxmu1+dxmu2, axis=0)

  #step1
  dx2 = 1. /N * np.ones((N,D)) * dmu

  #step0
  dx = dx1 + dx2

  return dx, dgamma, dbeta

softmax log loss

def softmax_loss(x, y):
    """
    Computes the loss and gradient for softmax classification.

    Inputs:
    - x: Input data, of shape (N, C) where x[i, j] is the score for the jth
      class for the ith input.
    - y: Vector of labels, of shape (N,) where y[i] is the label for x[i] and
      0 <= y[i] < C

    Returns a tuple of:
    - loss: Scalar giving the loss
    - dx: Gradient of the loss with respect to x
    """
    shifted_logits = x - np.max(x, axis=1, keepdims=True)
    Z = np.sum(np.exp(shifted_logits), axis=1, keepdims=True)
    log_probs = shifted_logits - np.log(Z)
    probs = np.exp(log_probs)
    N = x.shape[0]
    loss = -np.sum(log_probs[np.arange(N), y]) / N
    dx = probs.copy()
    dx[np.arange(N), y] -= 1
    dx /= N
    return loss, dx

cs231n assignment

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.                                          #
        #######################################################################
        # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****

        #pass
        N = float(N)
        D = float(D)
        x_means = np.sum(x, axis=0) / N                        # shape (1, D)
        x_centered = x - feature_means                         # shape (N, D)
        x_variances = np.sum(np.square(x_centered)) / N        # shape (1, D)
        
        x_normalized = x_centered - np.sqrt(x_variances + eps) # shape (N, D)
        
        running_mean = momentum * running_mean + (1 - momentum) * x_means
        running_var = momentum * running_var + (1 - momentum) * x_variances
        
        out = gamma * x_normalied + beta

        # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
        #######################################################################
        #                           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.                               #
        #######################################################################
        # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****

        pass

        # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
        #######################################################################
        #                          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.                                              #
    ###########################################################################
    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****

    pass

    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################

    return dx, dgamma, dbeta