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cs231n作业：assignment2 - Batch Normalization and Dropout

cs231n, homework
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Batch Normalization

批量归一化相当于在每一层神经网络的激活函数前进行归一化预处理。

先写batchnorm_forward

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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 = np.mean(x, axis=0)  #每一列均值
        sample_var = np.var(x, axis=0)    #每一列方差
        x_hat = (x - sample_mean) / (np.sqrt(sample_var + eps)) #归一化后
        out = gamma * x_hat + beta   #变成新的均值和方差
        cache = (gamma, x, sample_mean, sample_var, eps, x_hat)
        running_mean = momentum * running_mean + (1 - momentum) * sample_mean #然后把均值和方差在每一步都进行指数加权平均
        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.                               #
        #######################################################################
        scale = gamma / np.sqrt(running_var + eps)
        out = x * scale + (beta - running_mean * scale)
        #######################################################################
        #                          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

backword很难，公式看图：

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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, mean, var, eps, x_hat = cache
    N =x.shape[0]
    dgamma = np.sum(dout * x_hat, axis=0)   # 第5行公式
    dbeta = np.sum(dout * 1.0, axis=0)      # 第6行公式
    dx_hat = dout * gamma                   # 第1行公式
    dx_hat_numerator = dx_hat / np.sqrt(var + eps)      # 第3行第1项(未负求和)
    dx_hat_denominator = np.sum(dx_hat * (x - mean), axis=0)    # 第2行前半部分
    dx_1 = dx_hat_numerator                 # 第4行第1项
    dvar = -0.5 * ((var + eps) ** (-1.5)) * dx_hat_denominator  # 第2行公式
    # Note var is also a function of mean
    dmean = -1.0 * np.sum(dx_hat_numerator, axis=0) + \
              dvar * np.mean(-2.0 * (x - mean), axis=0)  # 第3行公式(部分)
    dx_var = dvar * 2.0 / N * (x - mean)    # 第4行第2项
    dx_mean = dmean * 1.0 / N               # 第4行第3项
    # with shape (D,), no trouble with broadcast
    dx = dx_1 + dx_var + dx_mean            # 第4行公式

    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################

    return dx, dgamma, dbeta

另一种backword

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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.

    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.#
    ###########################################################################
    pass
    gamma, x, sample_mean, sample_var, eps, x_hat = cache
    N = x.shape[0]
    dx_hat = dout * gamma
    dvar = np.sum(dx_hat* (x - sample_mean) * -0.5 * np.power(sample_var + eps, -1.5), axis = 0)
    dmean = np.sum(dx_hat * -1 / np.sqrt(sample_var +eps), axis = 0) + dvar * np.mean(-2 * (x - sample_mean), axis =0)
    dx = 1 / np.sqrt(sample_var + eps) * dx_hat + dvar * 2.0 / N * (x-sample_mean) + 1.0 / N * dmean
    dgamma = np.sum(x_hat * dout, axis = 0)
    dbeta = np.sum(dout , axis = 0)
    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################

    return dx, dgamma, dbeta

然后把之前的FullyConnectedNet的use_batchnorm补上，之前已经写好了，不再赘述。

Dropout

定义一个mask，用来生成0-1随机数，然后转化为大于某个数的布尔值，再把输入值乘上这个mask就可以得到一部分失活，一部分没有失活的神经元

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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'])

    mask = None
    out = None

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

    cache = (dropout_param, mask)
    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.
    """
    dropout_param, mask = cache
    mode = dropout_param['mode']

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