Liu K, Zuazua Iriondo E (2025)
Publication Type: Journal article, Original article
Publication year: 2025
Book Volume: 35
Pages Range: 1471-1521
Journal Issue: 6
URI: https://arxiv.org/html/2412.01619v1
DOI: 10.1142/S0218202525500228
In this work, we address three non-convex optimization problems associated with the training of shallow neural networks (NNs) for exact and approximate representation, as well as for regression tasks. Through a mean-field approach, we convexify these problems and, applying a representer theorem, prove the absence of relaxation gaps. We establish generalization bounds for the resulting NN solutions, assessing their predictive performance on test datasets, analyzing the impact of key hyperparameters on these bounds, and proposing optimal choices.
On the computational side, we examine the discretization of the convexified problems and derive convergence rates. For low-dimensional datasets, these discretized problems are efficiently solvable using the simplex method. For high-dimensional datasets, we propose a sparsification algorithm that, combined with gradient descent for over-parameterized shallow NNs, yields effective solutions to the primal problems.
APA:
Liu, K., & Zuazua Iriondo, E. (2025). Representation and Regression Problems in Neural Networks: Relaxation, Generalization, and Numerics. Mathematical Models & Methods in Applied Sciences, 35(6), 1471-1521. https://doi.org/10.1142/S0218202525500228
MLA:
Liu, Kang, and Enrique Zuazua Iriondo. "Representation and Regression Problems in Neural Networks: Relaxation, Generalization, and Numerics." Mathematical Models & Methods in Applied Sciences 35.6 (2025): 1471-1521.
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