Training Fully Connected Neural Networks Is $\exists\mathbb{R}$-Complete
Abstract
We consider the algorithmic problem of finding the optimal weights and biases for a two-layer fully connected neural network to fit a given set of data points, also known as empirical risk minimization. We show that the problem is $\exists\mathbb{R}$-complete. This complexity class can be defined as the set of algorithmic problems that are polynomial-time equivalent to finding real roots of a multivariate polynomial with integer coefficients. Furthermore, we show that arbitrary algebraic numbers are required as weights to be able to train some instances to optimality, even if all data points are rational. Our result already applies to fully connected instances with two inputs, two outputs, and one hidden layer of ReLU neurons. Thereby, we strengthen a result by Abrahamsen, Kleist and Miltzow [NeurIPS 2021]. A consequence of this is that a combinatorial search algorithm like the one by Arora, Basu, Mianjy and Mukherjee [ICLR 2018] is impossible for networks with more than one output dimension, unless $\text{NP} = \exists\mathbb{R}$.
Cite
Text
Bertschinger et al. "Training Fully Connected Neural Networks Is $\exists\mathbb{R}$-Complete." Neural Information Processing Systems, 2023.Markdown
[Bertschinger et al. "Training Fully Connected Neural Networks Is $\exists\mathbb{R}$-Complete." Neural Information Processing Systems, 2023.](https://mlanthology.org/neurips/2023/bertschinger2023neurips-training/)BibTeX
@inproceedings{bertschinger2023neurips-training,
title = {{Training Fully Connected Neural Networks Is $\exists\mathbb{R}$-Complete}},
author = {Bertschinger, Daniel and Hertrich, Christoph and Jungeblut, Paul and Miltzow, Tillmann and Weber, Simon},
booktitle = {Neural Information Processing Systems},
year = {2023},
url = {https://mlanthology.org/neurips/2023/bertschinger2023neurips-training/}
}