On Generalization Bounds for Neural Networks with Low Rank Layers

TL;DR

This paper analyzes how low-rank layers in deep neural networks can improve generalization bounds by preventing the accumulation of complexity factors, offering new insights into neural collapse phenomena.

Contribution

It introduces a novel analysis using Gaussian complexity to derive generalization bounds for low-rank constrained deep networks, highlighting their advantages over full-rank models.

Findings

Low-rank layers help prevent complexity accumulation across layers.

Networks with low-rank constraints can generalize better than full-rank networks.

The framework offers new perspectives on neural collapse and generalization.

Abstract

While previous optimization results have suggested that deep neural networks tend to favour low-rank weight matrices, the implications of this inductive bias on generalization bounds remain underexplored. In this paper, we apply Maurer's chain rule for Gaussian complexity to analyze how low-rank layers in deep networks can prevent the accumulation of rank and dimensionality factors that typically multiply across layers. This approach yields generalization bounds for rank and spectral norm constrained networks. We compare our results to prior generalization bounds for deep networks, highlighting how deep networks with low-rank layers can achieve better generalization than those with full-rank layers. Additionally, we discuss how this framework provides new perspectives on the generalization capabilities of deep networks exhibiting neural collapse.