On the numerical reliability of nonsmooth autodiff: a MaxPool case study
Ryan Boustany
This paper considers the reliability of automatic differentiation (AD) for
neural networks involving the nonsmooth MaxPool operation. We investigate the
behavior of AD across different precision levels (16, 32, 64 bits) and
convolutional architectures (LeNet, VGG, and ResNet) on various datasets
(MNIST, CIFAR10, SVHN, and ImageNet). Although AD can be incorrect, recent
research has shown that it coincides with the derivative almost everywhere,
even in the presence of nonsmooth operations (such as MaxPool and ReLU). On the
other hand, in practice, AD operates with floating-point numbers (not real
numbers), and there is, therefore, a need to explore subsets on which AD can be
numerically incorrect. These subsets include a bifurcation zone (where AD is
incorrect over reals) and a compensation zone (where AD is incorrect over
floating-point numbers but correct over reals). Using SGD for the training
process, we study the impact of different choices of the nonsmooth Jacobian for
the MaxPool function on the precision of 16 and 32 bits. These findings suggest
that nonsmooth MaxPool Jacobians with lower norms help maintain stable and
efficient test accuracy, whereas those with higher norms can result in
instability and decreased performance. We also observe that the influence of
MaxPool's nonsmooth Jacobians on learning can be reduced by using batch
normalization, Adam-like optimizers, or increasing the precision level.