# Training Neural Networks is existsmathbb R-complete

arXiv: https://arxiv.org/abs/2102.09798 · HTML: https://mlsanity.com/paper/2102.09798/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 53 references that are in the map.
- First posted on arXiv 19 Feb 2021.

## Builds on (most cited first, 11 of 11)

- [Training Deep Neural Networks with 8-bit Floating Point Numbers](https://mlsanity.com/paper/1812.08011/) (2018)
- [Optimal Continual Learning has Perfect Memory and is NP-hard](https://mlsanity.com/paper/2006.05188/) (2020)
- [Globally Optimal Gradient Descent for a ConvNet with Gaussian Inputs](https://mlsanity.com/paper/1702.07966/) (2017)
- [Tight Hardness Results for Training Depth-2 ReLU Networks](https://mlsanity.com/paper/2011.13550/) (2020)
- [Learning Two Layer Rectified Neural Networks in Polynomial Time](https://mlsanity.com/paper/1811.01885/) (2018)
- [Complexity of Training ReLU Neural Network](https://mlsanity.com/paper/1809.10787/) (2018)
- [Smoothing the gap between NP and ER](https://mlsanity.com/paper/1912.02278/) (2019)
- [Intersection Graphs of Rays and Grounded Segments](https://mlsanity.com/paper/1612.03638/) (2016)
- [The Complexity of Drawing a Graph in a Polygonal Region](https://mlsanity.com/paper/1802.06699/) (2018)
- [A Universality Theorem for Nested Polytopes](https://mlsanity.com/paper/1908.02213/) (2019)
- [Irrational Guards are Sometimes Needed](https://mlsanity.com/paper/1701.05475/) (2017)

## Abstract (arXiv)

Given a neural network, training data, and a threshold, it was known that it is NP-hard to find weights for the neural network such that the total error is below the threshold. We determine the algorithmic complexity of this fundamental problem precisely, by showing that it is existsmathbb R-complete. This means that the problem is equivalent, up to polynomial-time reductions, to deciding whether a system of polynomial equations and inequalities with integer coefficients and real unknowns has a solution. If, as widely expected, existsmathbb R is strictly larger than NP, our work implies that the problem of training neural networks is not even in NP. Neural networks are usually trained using some variation of backpropagation. The result of this paper offers an explanation why techniques commonly used to solve big instances of NP-complete problems seem not to be of use for this task. Examples of such techniques are SAT solvers, IP solvers, local search, dynamic programming, to name a few general ones.