Eigenspectrum Analysis of Neural Networks without Aspect Ratio Bias (2506.06280v1)

Abstract: Diagnosing deep neural networks (DNNs) through the eigenspectrum of weight matrices has been an active area of research in recent years. At a high level, eigenspectrum analysis of DNNs involves measuring the heavytailness of the empirical spectral densities (ESD) of weight matrices. It provides insight into how well a model is trained and can guide decisions on assigning better layer-wise training hyperparameters. In this paper, we address a challenge associated with such eigenspectrum methods: the impact of the aspect ratio of weight matrices on estimated heavytailness metrics. We demonstrate that matrices of varying sizes (and aspect ratios) introduce a non-negligible bias in estimating heavytailness metrics, 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 measuring the heavytailness of these submatrices with the fixed aspect ratio can effectively mitigate 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 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 these application domains. In one of the LLM pruning experiments, FARMS reduces the perplexity of the LLaMA-7B model by 17.3% when compared with the state-of-the-art method.

• The paper’s primary contribution is FARMS, which corrects the aspect ratio bias in weight matrix eigenspectrum analysis for more reliable training quality diagnostics.
• It employs fixed-aspect-ratio submatrix partitioning and averaging of ESDs to improve layer-wise hyperparameter tuning and error reduction across diverse neural network architectures.
• Empirical validation demonstrates FARMS’ impact, reducing perplexity in LLMs and stabilizing training in image classification and SciML applications.

Eigenspectrum Analysis of Neural Networks without Aspect Ratio Bias

The paper "Eigenspectrum Analysis of Neural Networks without Aspect Ratio Bias" by Yuanzhe Hu et al. addresses a critical issue in the spectral analysis of deep neural networks (DNNs): the aspect ratio bias in weight matrix eigenspectrum analysis. The authors propose a novel method, FARMS (Fixed-Aspect-Ratio Matrix Subsampling), to mitigate this bias and enhance the accuracy of training quality assessment in various neural network applications.

Background and Motivation

Eigenspectrum analysis is a potent tool for diagnosing DNNs, providing insights into training dynamics by evaluating the heavytailness of empirical spectral densities (ESDs) of weight matrices. This approach, grounded in Heavy-Tailed Self-Regularization (HT-SR) theory, correlates the spectral properties of these matrices with training quality. However, a known limitation arises from the aspect ratio of weight matrices, which can skew the analysis, leading to inaccurate assessments. This bias has implications for layer-wise hyperparameter tuning, such as learning rates and pruning ratios.

The authors identify that variations in matrix aspect ratios can artificially alter the ESDs. Conventional methods neglect this dependency, resulting in potential misdiagnosis of model layer quality, especially in architectures with significant discrepancies in layer dimensions.

FARMS: The Proposed Solution

FARMS addresses the aspect ratio bias by subsampling fixed-aspect-ratio submatrices from the original matrices. The process involves:

1. Partitioning weight matrices into overlapping submatrices, maintaining a uniform aspect ratio across all layers.
2. Averaging the ESDs of these submatrices to compute HT metrics, thus yielding a more robust measure of training quality regardless of the original matrix size.

This method ensures that spectral analysis reflects genuine training characteristics, not artifacts of matrix geometry.

Empirical Validation

The paper validates FARMS across multiple domains, including computer vision, scientific machine learning, and LLM pruning. Key findings:

• LLMs: With FARMS, the perplexity of the LLaMA-7B model is reduced by 17.3% compared to state-of-the-art pruning methods. Such improvements illustrate the method's efficacy in practical applications.
• Image Classification: FARMS enhances layer-wise learning rate assignments in ResNet and VGG architectures, improving test accuracy and stabilizing training across various layer dimension configurations.
• Scientific Machine Learning (SciML): FARMS aids in model fine-tuning, achieving notable error reductions across diverse datasets.

Implications and Future Work

FARMS presents a significant advancement in spectral analysis, with implications for model diagnostics and layer-wise optimization processes. By providing a more accurate reflection of training dynamics, this method can lead to better-informed decisions in model training and architecture design.

Theoretically, this work suggests new possibilities for exploring the interplay between matrix geometry and spectral properties in neural networks. Practically, it opens pathways for developing more sophisticated training and pruning strategies that leverage a nuanced understanding of a model's spectral characteristics.

Future research could expand on calibrating subsampling parameters and explore its integration with other optimization methods like adversarial training or knowledge distillation. Additionally, further exploration into its applicability across different neural network architectures and scaling behaviors could enhance the versatility of this approach.

In conclusion, FARMS marks a notable step toward refining neural network analysis by neutralizing the aspect ratio bias in spectral diagnostics, thus equipping researchers and practitioners with a more precise tool for evaluating and improving model training quality.