Eigenspectrum Analysis of Neural Networks without Aspect Ratio Bias

Abstract

Diagnosing deep neural networks (DNNs) by analyzing the eigenspectrum of their weights has been an active area of research in recent years. One of the main approaches involves measuring the heavytailness of the empirical spectral densities (ESDs) of weight matrices. This analysis has been shown to provide insights to help diagnose whether a model is well-trained or undertrained, and has been used to guide training methods involving layer-wise hyperparameter assignment. In this paper, we address an often-overlooked challenge in estimating the heavytailness of these ESDs: the impact of the aspect ratio of weight matrices. We demonstrate that matrices of varying sizes (and aspect ratios) introduce a non-negligible bias in estimating the heavytailness of ESDs, leading to inaccurate model diagnosis and layer-wise hyperparameter assignment. To overcome this challenge, we propose FARMS (Fixed-Aspect-Ratio Matrix Subsampling), a method that normalizes the weight matrices by subsampling submatrices with a fixed aspect ratio. Instead of measuring the heavytailness of the original ESD, we measure the average ESD of these subsampled submatrices. We show that this method effectively mitigates the aspect ratio bias. We validate our approach across various optimization techniques and application domains that involve eigenspectrum analysis of weights, including image classification in computer vision (CV) models, scientific machine learning (SciML) model training, and large language model (LLM) pruning. Our results show that despite its simplicity, FARMS uniformly improves the accuracy of eigenspectrum analysis while enabling more effective layer-wise hyperparameter assignment. In one of the LLM pruning experiments, FARMS reduces the perplexity of the LLaMA-7B model by 17.3% when compared with state-of-the-art methods.