On the Correctness of Automatic Differentiation for Neural Networks with Machine-Representable Parameters

Wonyeol Lee, Sejun Park, Alex Aiken
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:19094-19140, 2023.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-lee23p, title = {On the Correctness of Automatic Differentiation for Neural Networks with Machine-Representable Parameters}, author = {Lee, Wonyeol and Park, Sejun and Aiken, Alex}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {19094--19140}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/lee23p/lee23p.pdf}, url = {https://proceedings.mlr.press/v202/lee23p.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T On the Correctness of Automatic Differentiation for Neural Networks with Machine-Representable Parameters %A Wonyeol Lee %A Sejun Park %A Alex Aiken %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-lee23p %I PMLR %P 19094--19140 %U https://proceedings.mlr.press/v202/lee23p.html %V 202 %X 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.
APA
Lee, W., Park, S. & Aiken, A.. (2023). On the Correctness of Automatic Differentiation for Neural Networks with Machine-Representable Parameters. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:19094-19140 Available from https://proceedings.mlr.press/v202/lee23p.html.

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