Is "chainer.functions.sigmoid_cross_entropy" a second-order differentiable function?

Question

I am a student studying machine learning. For my study, we need to differentiate the loss function by second order, we use "chainer.functions.sigmoid_cross_entropy". A similar function is "chainer.functions.softmax_cross_entropy". This function has an argument ", enable_double_backprop" to realize the second derivative, but not in "chainer.functions.sigmoid_cross_entropy".

Is "chainer.functions.sigmoid_cross_entropy" a second-order differentiable function?

Please teach me!

chainer.functions.sigmoid_cross_entropy (x, t, normalize = True, reduce = 'mean')

chainer.functions.softmax_cross_entropy (x, t, normalize = True, cache_score = True, 
class_weight = None, ignore_label = -1, reduce = 'mean', enable_double_backprop = False, 
soft_target_loss = 'cross-entropy')

Answer 1

Yes, sigmoid_cross_entropy is second-order differentiable. For performance reasons, softmax_cross_entropy is not second-order differentiable unless enable_double_backprop=True is given.

Functions that does not support higher-order derivatives are listed in https://github.com/chainer/chainer/issues/4449.