$K$-MSHC: Unmasking Minimally Sufficient Head Circuits in Large Language Models with Experiments on Syntactic Classification Tasks

TL;DR

This paper introduces a method to identify minimal attention head circuits in large language models for specific tasks, revealing task-specific and partially overlapping neural components in syntactic and arithmetic tasks.

Contribution

The paper proposes the $(m{K}, oldsymbol{ ext{epsilon}})$-Minimum Sufficient Head Circuit ($K$-MSHC) methodology and the Search-K-MSHC algorithm to uncover minimal task-critical attention heads in large language models.

Findings

Distinct task-specific head circuits identified

Early layers dominate grammar tasks, while word problems involve multiple layers

Shared weak heads exist between grammar and arithmetic, but strong heads are task-specific

Abstract

Understanding which neural components drive specific capabilities in mid-sized language models ($\leq$10B parameters) remains a key challenge. We introduce the $(\bm{K}, \epsilon)$-Minimum Sufficient Head Circuit ($K$-MSHC), a methodology to identify minimal sets of attention heads crucial for classification tasks as well as Search-K-MSHC, an efficient algorithm for discovering these circuits. Applying our Search-K-MSHC algorithm to Gemma-9B, we analyze three syntactic task families: grammar acceptability, arithmetic verification, and arithmetic word problems. Our findings reveal distinct task-specific head circuits, with grammar tasks predominantly utilizing early layers, word problems showing pronounced activity in both shallow and deep regions, and arithmetic verification demonstrating a more distributed pattern across the network. We discover non-linear circuit overlap patterns, where different task pairs share computational components at varying levels of importance. While grammar and arithmetic share many "weak" heads, arithmetic and word problems share more consistently critical "strong" heads. Importantly, we find that each task maintains dedicated "super-heads" with minimal cross-task overlap, suggesting that syntactic and numerical competencies emerge from specialized yet partially reusable head circuits.