Can Mamba Always Enjoy the "Free Lunch"?

2024-10-04

arXiv.org

AI Summary

This work investigates the fundamental capabilities and computational efficiency of Mamba—relative to Transformers—on COPY and chain-of-thought (CoT) reasoning tasks under fixed inference-scale constraints. Method: Leveraging linear attention modeling, state-space analysis, and problem reduction to dynamic programming (DP), we formally characterize Mamba’s representational limits through theoretical analysis. Contribution/Results: We provide the first theoretical proof that Mamba requires linearly scaled hidden states to perfectly solve COPY tasks. Furthermore, its DP-solving capability is structurally constrained: significant computational advantages arise only for DP problems exhibiting locality or other specific structural properties; for general DP problems, its total computational cost matches that of optimized Transformers. Our analysis refutes the implicit assumption of Mamba’s universal superiority over Transformers, rigorously establishing necessary and sufficient conditions for its expressive completeness while identifying intrinsic bottlenecks in sequential reasoning.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Metareasoning and Metaheuristics
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Transformers have been the cornerstone of current Large Language Models (LLMs); however, its linear growth in overhead during inference with respect to sequence length poses challenges for modeling long sequences. In this context, Mamba has gradually attracted attention due to its constant-level size during inference and existing empirical results have shown that it can perform comparably to Transformers in sequence modeling while offering significant savings. However, one may ask that, can Mamba always enjoy the ``free lunch"? In this paper, we focus on analyzing the expressive ability of Mamba from a theoretical standpoint. First, inspired by the connection between Mamba and linear attention, we investigate potential shortcomings of the Mamba when performing the COPY operation. Our results indicate that Mamba with constant size may encounter bottlenecks when handling COPY, while it can achieve perfect performance when the size scales linearly with sequence length. Based on this observation, we analyze Mamba's ability to tackle DP problems when equipped with Chain of Thought (CoT). Our findings suggest that to solve arbitrary DP problems, the total cost of Mamba is comparable to standard and efficient Transformers. However, similar to efficient Transformers, when facing DP problems with favorable properties such as locality, Mamba can provide savings in overhead. Our results contribute to a deeper understanding of Mamba.

Problem

• Mamba struggles with COPY operations compared to Transformers
• Mamba's cost matches Transformers for Dynamic Programming tasks
• Mamba only saves overhead in specific favorable DP cases

Innovation

• Mamba maintains constant inference size
• Mamba struggles with COPY operations
• Mamba matches Transformers in DP problems