Controllability Analysis of State Space-based Language Model

TL;DR

This paper introduces the Influence Score, a controllability metric for state-space models like Mamba, revealing how internal dynamics relate to model size, architecture, and emergent behaviors, and providing a new diagnostic tool.

Contribution

The paper proposes and validates the Influence Score, a novel controllability metric for state-space language models, enabling better understanding and comparison of their internal dynamics.

Findings

Influence Score increases with model size and training data.

Mamba models show recency bias and influence concentrated in mid-to-late layers.

Emergent behaviors only appear at larger scales, affecting token influence patterns.

Abstract

State-space models (SSMs), particularly Mamba, have become powerful architectures for sequence modeling, yet their internal dynamics remain poorly understood compared to attention-based models. We introduce and validate the Influence Score, a controllability-based metric derived from the discretized state-space parameters of Mamba and computed through a backward recurrence analogous to system observability. The score quantifies how strongly a token at position k affects all later states and outputs. We evaluate this measure across three Mamba variants: mamba-130m, mamba-2.8b, and mamba-2.8b-slimpj, using six experiments that test its sensitivity to temperature, prompt complexity, token type, layer depth, token position, and input perturbations. The results show three main insights: (1) the Influence Score increases with model size and training data, reflecting model capacity; (2) Mamba exhibits consistent architectural patterns, including recency bias and concentrated influence in mid-to-late layers; and (3) emergent behaviors appear only at scale, with mamba-2.8b-slimpj uniquely prioritizing content words and reducing internal influence in the presence of noise. These findings establish the Influence Score as a practical diagnostic tool for interpreting and comparing SSM-based language models.