A Convex Relaxation Approach to Generalization Analysis for Parallel Positively Homogeneous Networks

TL;DR

This paper introduces a convex relaxation framework for analyzing the generalization of parallel positively homogeneous neural networks, providing bounds that scale nearly linearly with network width across various models.

Contribution

It develops a unified convex relaxation approach for deriving generalization bounds applicable to a broad class of positively homogeneous neural networks, including novel models like multi-head attention.

Findings

Generalization bounds scale almost linearly with network width.

Framework applies to diverse models including matrix sensing and attention mechanisms.

Provides a global lower-bound linking non-convex ERM to convex optimization.

Abstract

We propose a general framework for deriving generalization bounds for parallel positively homogeneous neural networks--a class of neural networks whose input-output map decomposes as the sum of positively homogeneous maps. Examples of such networks include matrix factorization and sensing, single-layer multi-head attention mechanisms, tensor factorization, deep linear and ReLU networks, and more. Our general framework is based on linking the non-convex empirical risk minimization (ERM) problem to a closely related convex optimization problem over prediction functions, which provides a global, achievable lower-bound to the ERM problem. We exploit this convex lower-bound to perform generalization analysis in the convex space while controlling the discrepancy between the convex model and its non-convex counterpart. We apply our general framework to a wide variety of models ranging from low-rank matrix sensing, to structured matrix sensing, two-layer linear networks, two-layer ReLU networks, and single-layer multi-head attention mechanisms, achieving generalization bounds with a sample complexity that scales almost linearly with the network width.