Sparsing Law: Towards Large Language Models with Greater Activation Sparsity

TL;DR

This paper investigates the factors influencing activation sparsity in large language models, proposing a new metric and revealing empirical laws that can guide the development of more efficient and interpretable LLMs.

Contribution

It introduces PPL-$p ext{ extbackslash}%$ sparsity, a novel quantitative metric for activation sparsity applicable to any activation function, and provides comprehensive empirical insights into sparsity scaling laws.

Findings

ReLU achieves higher activation sparsity than SiLU with more training data.

Activation ratio follows power-law and logspace power-law growth with training data for SiLU and ReLU.

Deeper architectures at fixed parameters can enhance activation sparsity.

Abstract

Activation sparsity denotes the existence of substantial weakly-contributed elements within activation outputs that can be eliminated, benefiting many important applications concerned with large language models (LLMs). Although promoting greater activation sparsity within LLMs deserves deep studies, existing works lack comprehensive and quantitative research on the correlation between activation sparsity and potentially influential factors. In this paper, we present a comprehensive study on the quantitative scaling properties and influential factors of the activation sparsity within decoder-only Transformer-based LLMs. Specifically, we propose PPL-$p\%$ sparsity, a precise and performance-aware activation sparsity metric that is applicable to any activation function. Through extensive experiments, we find several important phenomena. Firstly, different activation functions exhibit comparable performance but opposite training-time sparsity trends. The activation ratio (i.e., $1-\mathrm{sparsity\ ratio}$) evolves as a convergent increasing power-law and decreasing logspace power-law with the amount of training data for SiLU-activated and ReLU-activated LLMs, respectively. These demonstrate that ReLU is more efficient as the activation function than SiLU and can leverage more training data to improve activation sparsity. Secondly, the activation ratio linearly increases with the width-depth ratio below a certain bottleneck point, indicating the potential advantage of a deeper architecture at a fixed parameter scale. Finally, at similar width-depth ratios, we surprisingly find that the limit value of activation sparsity varies weakly with the parameter scale, i.e., the activation patterns within LLMs are insensitive to the parameter scale. These empirical laws towards LLMs with greater activation sparsity have important implications for making LLMs more efficient and interpretable.