  Recent work has shown that forward- and reverse- mode automatic
differentiation (AD) over the reals is almost always correct in a
mathematically precise sense. However, actual programs work with
machine-representable numbers (e.g., floating-point numbers), not reals. In
this paper, we study the correctness of AD when the parameter space of a neural
network consists solely of machine-representable numbers. In particular, we
analyze two sets of parameters on which AD can be incorrect: the incorrect set
on which the network is differentiable but AD does not compute its derivative,
and the non-differentiable set on which the network is non-differentiable. For
a neural network with bias parameters, we first prove that the incorrect set is
always empty. We then prove a tight bound on the size of the non-differentiable
set, which is linear in the number of non-differentiabilities in activation
functions, and give a simple necessary and sufficient condition for a parameter
to be in this set. We further prove that AD always computes a Clarke
subderivative even on the non-differentiable set. We also extend these results
to neural networks possibly without bias parameters.

Machine Learning;Machine Learning