Convex Formulations for Training Two-Layer ReLU Neural Networks

TL;DR

This paper introduces a convex reformulation of training infinite-width two-layer ReLU neural networks, enabling polynomial-time approximations and demonstrating competitive test accuracy in classification tasks.

Contribution

It presents the first convex formulation for training two-layer ReLU networks and proposes a semidefinite relaxation to make the problem computationally feasible.

Findings

Semidefinite relaxation is effective and can be solved in polynomial time.

The relaxation achieves competitive test accuracy on various classification tasks.

The convex formulation provides insights into neural network training dynamics.

Abstract

Solving non-convex, NP-hard optimization problems is crucial for training machine learning models, including neural networks. However, non-convexity often leads to black-box machine learning models with unclear inner workings. While convex formulations have been used for verifying neural network robustness, their application to training neural networks remains less explored. In response to this challenge, we reformulate the problem of training infinite-width two-layer ReLU networks as a convex completely positive program in a finite-dimensional (lifted) space. Despite the convexity, solving this problem remains NP-hard due to the complete positivity constraint. To overcome this challenge, we introduce a semidefinite relaxation that can be solved in polynomial time. We then experimentally evaluate the tightness of this relaxation, demonstrating its competitive performance in test accuracy across a range of classification tasks.