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

TL;DR

This paper develops a convexified framework for training shallow neural networks addressing representation and regression problems, providing theoretical guarantees, generalization bounds, and efficient algorithms for different data dimensions.

Contribution

It introduces a mean-field convexification approach, proves the absence of relaxation gaps, and proposes scalable algorithms for high-dimensional data.

Findings

Convexification via mean-field approach eliminates relaxation gaps.

Generalization bounds depend on key hyperparameters.

Efficient algorithms are proposed for low- and high-dimensional datasets.

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.