# On the numerical reliability of nonsmooth autodiff: a MaxPool case study

arXiv: https://arxiv.org/abs/2401.02736 · HTML: https://mlsanity.com/paper/2401.02736/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 45 references that are in the map.
- First posted on arXiv 5 Jan 2024.

## Builds on (most cited first, 21 of 21)

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- [Nonsmooth Implicit Differentiation for Machine Learning and Optimization](https://mlsanity.com/paper/2106.04350/) (2021)
- [On Correctness of Automatic Differentiation for Non-Differentiable Functions](https://mlsanity.com/paper/2006.06903/) (2020)
- [Provably Correct Automatic Subdifferentiation for Qualified Programs](https://mlsanity.com/paper/1809.08530/) (2018)
- [On the complexity of nonsmooth automatic differentiation](https://mlsanity.com/paper/2206.01730/) (2022)
- [On the Correctness of Automatic Differentiation for Neural Networks with Machine-Representable Parameters](https://mlsanity.com/paper/2301.13370/) (2023)

## Abstract (arXiv)

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.