本文研究Keras自带的几个常用的Loss Function。

1. categorical_crossentropy VS. sparse_categorical_crossentropy

注意到二者的主要差别在于输入是否为integer tensor。在文档中，我们还可以找到关于二者如何选择的描述：

解释一下这里的Integer target 与 Categorical target，实际上Integer target经过独热编码就变成了Categorical target，举例说明：

(类别数5)
Integer target: [1,2,4]
Categorical target： [[0. 1. 0. 0. 0.]
					 [0. 0. 1. 0. 0.]
					 [0. 0. 0. 0. 1.]]

在Keras中提供了to_categorical方法来实现二者的转化：

from keras.utils import to_categorical
categorical_labels = to_categorical(int_labels, num_classes=None)

注意categorical_crossentropy和sparse_categorical_crossentropy的输入参数output，都是softmax输出的tensor。我们都知道softmax的输出服从多项分布，

因此categorical_crossentropy和sparse_categorical_crossentropy应当应用于多分类问题。

我们再看看这两个的源码，来验证一下：

https://github.com/tensorflow/tensorflow/blob/r1.13/tensorflow/python/keras/backend.py
--------------------------------------------------------------------------------------------------------------------
def categorical_crossentropy(target, output, from_logits=False, axis=-1):
  """Categorical crossentropy between an output tensor and a target tensor.
  Arguments:
      target: A tensor of the same shape as `output`.
      output: A tensor resulting from a softmax
          (unless `from_logits` is True, in which
          case `output` is expected to be the logits).
      from_logits: Boolean, whether `output` is the
          result of a softmax, or is a tensor of logits.
      axis: Int specifying the channels axis. `axis=-1` corresponds to data
          format `channels_last', and `axis=1` corresponds to data format
          `channels_first`.
  Returns:
      Output tensor.
  Raises:
      ValueError: if `axis` is neither -1 nor one of the axes of `output`.
  """
  rank = len(output.shape)
  axis = axis % rank
  # Note: nn.softmax_cross_entropy_with_logits_v2
  # expects logits, Keras expects probabilities.
  if not from_logits:
    # scale preds so that the class probas of each sample sum to 1
    output = output / math_ops.reduce_sum(output, axis, True)
    # manual computation of crossentropy
    epsilon_ = _to_tensor(epsilon(), output.dtype.base_dtype)
    output = clip_ops.clip_by_value(output, epsilon_, 1. - epsilon_)
    return -math_ops.reduce_sum(target * math_ops.log(output), axis)
  else:
    return nn.softmax_cross_entropy_with_logits_v2(labels=target, logits=output)
--------------------------------------------------------------------------------------------------------------------
def sparse_categorical_crossentropy(target, output, from_logits=False, axis=-1):
  """Categorical crossentropy with integer targets.
  Arguments:
      target: An integer tensor.
      output: A tensor resulting from a softmax
          (unless `from_logits` is True, in which
          case `output` is expected to be the logits).
      from_logits: Boolean, whether `output` is the
          result of a softmax, or is a tensor of logits.
      axis: Int specifying the channels axis. `axis=-1` corresponds to data
          format `channels_last', and `axis=1` corresponds to data format
          `channels_first`.
  Returns:
      Output tensor.
  Raises:
      ValueError: if `axis` is neither -1 nor one of the axes of `output`.
  """
  rank = len(output.shape)
  axis = axis % rank
  if axis != rank - 1:
    permutation = list(range(axis)) + list(range(axis + 1, rank)) + [axis]
    output = array_ops.transpose(output, perm=permutation)
  # Note: nn.sparse_softmax_cross_entropy_with_logits
  # expects logits, Keras expects probabilities.
  if not from_logits:
    epsilon_ = _to_tensor(epsilon(), output.dtype.base_dtype)
    output = clip_ops.clip_by_value(output, epsilon_, 1 - epsilon_)
    output = math_ops.log(output)
  output_shape = output.shape
  targets = cast(flatten(target), 'int64')
  logits = array_ops.reshape(output, [-1, int(output_shape[-1])])
  res = nn.sparse_softmax_cross_entropy_with_logits(
      labels=targets, logits=logits)
  if len(output_shape) >= 3:
    # If our output includes timesteps or spatial dimensions we need to reshape
    return array_ops.reshape(res, array_ops.shape(output)[:-1])
  else:
    return res

categorical_crossentropy计算交叉熵时使用的是nn.softmax_cross_entropy_with_logits_v2( labels=targets, logits=logits)，而sparse_categorical_crossentropy使用的是nn.sparse_softmax_cross_entropy_with_logits( labels=targets, logits=logits)，二者本质并无区别，只是对输入参数logits的要求不同，v2要求的是logits与labels格式相同（即元素也是独热的），而sparse则要求logits的元素是个数值，与上面Integer format和Categorical format的对比含义类似。

综上所述，categorical_crossentropy和sparse_categorical_crossentropy只不过是输入参数target类型上的区别，其loss的计算在本质上没有区别，就是交叉熵；二者是针对多分类（Multi-class）任务的。

2. Binary_crossentropy

二元交叉熵，从名字中我们可以看出，这个loss function可能是适用于二分类的。文档中并没有详细说明，那么直接看看源码吧：

https://github.com/tensorflow/tensorflow/blob/r1.13/tensorflow/python/keras/backend.py
--------------------------------------------------------------------------------------------------------------------
def binary_crossentropy(target, output, from_logits=False):
  """Binary crossentropy between an output tensor and a target tensor.
  Arguments:
      target: A tensor with the same shape as `output`.
      output: A tensor.
      from_logits: Whether `output` is expected to be a logits tensor.
          By default, we consider that `output`
          encodes a probability distribution.
  Returns:
      A tensor.
  """
  # Note: nn.sigmoid_cross_entropy_with_logits
  # expects logits, Keras expects probabilities.
  if not from_logits:
    # transform back to logits
    epsilon_ = _to_tensor(epsilon(), output.dtype.base_dtype)
    output = clip_ops.clip_by_value(output, epsilon_, 1 - epsilon_)
    output = math_ops.log(output / (1 - output))
  return nn.sigmoid_cross_entropy_with_logits(labels=target, logits=output)

可以看到源码中计算使用了nn.sigmoid_cross_entropy_with_logits，熟悉tensorflow的应该比较熟悉这个损失函数了，它可以用于简单的二分类，也可以用于多标签任务，而且应用广泛，在样本合理的情况下（如不存在类别不均衡等问题）的情况下，通常可以直接使用。

补充：keras自定义loss function的简单方法

首先看一下Keras中我们常用到的目标函数（如mse,mae等）是如何定义的

from keras import backend as K
def mean_squared_error(y_true, y_pred):
    return K.mean(K.square(y_pred - y_true), axis=-1)
def mean_absolute_error(y_true, y_pred):
    return K.mean(K.abs(y_pred - y_true), axis=-1)
def mean_absolute_percentage_error(y_true, y_pred):
    diff = K.abs((y_true - y_pred) / K.clip(K.abs(y_true), K.epsilon(), np.inf))
    return 100. * K.mean(diff, axis=-1)
def categorical_crossentropy(y_true, y_pred):
    '''Expects a binary class matrix instead of a vector of scalar classes.
    '''
    return K.categorical_crossentropy(y_pred, y_true)
def sparse_categorical_crossentropy(y_true, y_pred):
    '''expects an array of integer classes.
    Note: labels shape must have the same number of dimensions as output shape.
    If you get a shape error, add a length-1 dimension to labels.
    '''
    return K.sparse_categorical_crossentropy(y_pred, y_true)
def binary_crossentropy(y_true, y_pred):
    return K.mean(K.binary_crossentropy(y_pred, y_true), axis=-1)
def kullback_leibler_divergence(y_true, y_pred):
    y_true = K.clip(y_true, K.epsilon(), 1)
    y_pred = K.clip(y_pred, K.epsilon(), 1)
    return K.sum(y_true * K.log(y_true / y_pred), axis=-1)
def poisson(y_true, y_pred):
    return K.mean(y_pred - y_true * K.log(y_pred + K.epsilon()), axis=-1)
def cosine_proximity(y_true, y_pred):
    y_true = K.l2_normalize(y_true, axis=-1)
    y_pred = K.l2_normalize(y_pred, axis=-1)
    return -K.mean(y_true * y_pred, axis=-1)

所以仿照以上的方法，可以自己定义特定任务的目标函数。比如：定义预测值与真实值的差

from keras import backend as K
def new_loss(y_true,y_pred):
    return K.mean((y_pred-y_true),axis = -1)

然后，应用你自己定义的目标函数进行编译

from keras import backend as K
def my_loss(y_true,y_pred):
    return K.mean((y_pred-y_true),axis = -1)
model.compile(optimizer=optimizers.RMSprop(lr)，loss=my_loss,
metrics=['accuracy'])

以上为个人经验，希望能给大家一个参考，也希望大家多多支持hwidc。

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