本文研究Keras自带的几个常用的Loss Functions。
1. categorical_crossentropy VS. sparse_categorical_crossentropy
先看categorical_crossentropy和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
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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)
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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形式上的区别，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
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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。