Constant-size ReLU networks with hidden out-degree 1 and linear networks admitting an 'untangling' are optimizable in polynomial time, via hyperplane-partition enumeration and constrained linear regression.
Training Neural Networks is NP-Hard in Fixed Dimension
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abstract
We study the parameterized complexity of training two-layer neural networks with respect to the dimension of the input data and the number of hidden neurons, considering ReLU and linear threshold activation functions. Albeit the computational complexity of these problems has been studied numerous times in recent years, several questions are still open. We answer questions by Arora et al. [ICLR '18] and Khalife and Basu [IPCO '22] showing that both problems are NP-hard for two dimensions, which excludes any polynomial-time algorithm for constant dimension. We also answer a question by Froese et al. [JAIR '22] proving W[1]-hardness for four ReLUs (or two linear threshold neurons) with zero training error. Finally, in the ReLU case, we show fixed-parameter tractability for the combined parameter number of dimensions and number of ReLUs if the network is assumed to compute a convex map. Our results settle the complexity status regarding these parameters almost completely.
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cs.LG 1years
2026 1verdicts
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New Complexity-Theoretic Frontiers of Tractability for Neural Network Training
Constant-size ReLU networks with hidden out-degree 1 and linear networks admitting an 'untangling' are optimizable in polynomial time, via hyperplane-partition enumeration and constrained linear regression.