Training Neural Networks is $\exists\mathbb R$-complete

TL;DR

This paper proves that training neural networks is $orall ext{R}$-complete, establishing its precise computational complexity and explaining why common NP-hard problem-solving techniques are ineffective.

Contribution

It establishes that training neural networks is $orall ext{R}$-complete, a complexity class beyond NP, providing a fundamental complexity classification for this problem.

Findings

Training neural network weights is $orall ext{R}$-complete.

The problem is equivalent to solving polynomial systems with real unknowns.

Common NP-hard problem-solving techniques are ineffective for neural network training.

Abstract

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 $\exists\mathbb R$-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, $\exists\mathbb R$ 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.