Representation and Regression Problems in Neural Networks: Relaxation, Generalization, and Numerics

• URL: http://arxiv.org/abs/2412.01619v1
• Date: Mon, 02 Dec 2024 15:40:29 GMT
• Title: Representation and Regression Problems in Neural Networks: Relaxation, Generalization, and Numerics
• Authors: Kang Liu, Enrique Zuazua,
• Abstract summary: We address three non-dimensional optimization problems associated with training shallow neural networks (NNs) We convexify these problems and representation, applying a representer gradient to prove the absence relaxation gaps. We analyze the impact of key parameters on these bounds, propose optimal choices. For high-dimensional datasets, we propose a sparsification algorithm that, combined with gradient descent, yields effective solutions.
• Score: 5.915970073098098
• License:
• Abstract: 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 and, analyzing the impact of key hyperparameters on these bounds, propose 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.

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