Large Language Model Evaluation via Matrix Nuclear-Norm

2024-10-14

arXiv.org

AI Summary

To address the high computational complexity (O(n³)) and poor scalability of matrix entropy—commonly used to evaluate information compression capability in large language models (LLMs)—this work proposes a lightweight evaluation metric based on nuclear norm. We introduce the nuclear norm as the first convex surrogate for quantifying LLM compression capacity and design an efficient L_{1,2}-norm approximation scheme that entirely bypasses singular value decomposition (SVD), thereby ensuring both theoretical interpretability and engineering scalability. Evaluated on Cerebras-GPT (111M–6.7B) and Pythia models, our method achieves 8–24× speedup over matrix entropy. It maintains strong rank-order consistency with matrix entropy across multiple benchmarks and generative response assessments (Spearman ρ > 0.92), significantly enhancing the efficiency and practicality of assessing representational diversity and redundancy in large-scale models.

Technology Category

• Machine Learning: Large Multimodal Models (LMMs)
• Natural Language Processing: (Large) Language Models
• Computer Vision: Large Vision Models

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

As large language models (LLMs) continue to evolve, efficient evaluation metrics are vital for assessing their ability to compress information and reduce redundancy. While traditional metrics like Matrix Entropy offer valuable insights, they are computationally intensive for large-scale models due to their ( O(n^3) ) time complexity with Singular Value Decomposition (SVD). To mitigate this issue, we introduce the Matrix Nuclear-Norm, which not only serves as a metric to quantify the data compression proficiency of LLM but also provides a convex approximation of matrix rank to capture both predictive discriminability and diversity. By employing the ( L_{1,2} ext{-norm} ) to further approximate the nuclear norm, we can effectively assess the model's information compression capabilities. This approach reduces the time complexity to ( O(n^2) ) and eliminates the need for SVD computation. Consequently, the Matrix Nuclear-Norm achieves speeds 8 to 24 times faster than Matrix Entropy for the CEREBRAS-GPT model as sizes increase from 111M to 6.7B. This performance gap becomes more pronounced with larger models, as validated in tests with other models like Pythia. Additionally, evaluations on benchmarks and model responses confirm that our proposed Matrix Nuclear-Norm is a reliable, scalable, and efficient tool for assessing LLMs' performance, striking a balance between accuracy and computational efficiency. The code is available at https://github.com/MLGroupJLU/MatrixNuclearNorm.

Problem

• Efficiently evaluate large language models' information compression ability
• Reduce computational complexity of traditional matrix entropy metrics
• Provide scalable convex approximation for model performance assessment

Innovation

• Matrix Nuclear-Norm replaces SVD for efficiency
• L1,2-norm approximates nuclear norm effectively
• Reduces time complexity to O(n^2)