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Training Fully Connected Neural Networks is $\exists\mathbb{R}$-Complete
We consider the problem of finding weights and biases for a two-layer fully connected neural network to fit a given set of data points as well as possible, also known as EmpiricalRiskMinimization.
On Classifying Continuous Constraint Satisfaction Problems
In an instance of this problem we are given some sentence of the form $\exists x_1, \ldots, x_n \in \mathbb{R} : \Phi(x_1, \ldots, x_n)$, where $\Phi$ is a well-formed quantifier-free formula consisting of the symbols $\{0, 1, +, \cdot, \geq, >, \wedge, \vee, \neg\}$, the goal is to check whether this sentence is true.
Training Neural Networks is ER-complete
Given a neural network, training data, and a threshold, finding weights for the neural network such that the total error is below the threshold is known to be NP-hard.
Training Neural Networks is $\exists\mathbb R$-complete
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.
On the VC-dimension of half-spaces with respect to convex sets
A family S of convex sets in the plane defines a hypergraph H = (S, E) as follows.
Computational Geometry Discrete Mathematics Data Structures and Algorithms Combinatorics
