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The Expressive Capacity of State Space Models: A Formal Language Perspective (2405.17394v2)

Published 27 May 2024 in cs.CL, cs.FL, and cs.LG

Abstract: Recently, recurrent models based on linear state space models (SSMs) have shown promising performance in LLMing (LM), competititve with transformers. However, there is little understanding of the in-principle abilities of such models, which could provide useful guidance to the search for better LM architectures. We present a comprehensive theoretical study of the capacity of such SSMs as it compares to that of transformers and traditional RNNs. We find that SSMs and transformers have overlapping but distinct strengths. In star-free state tracking, SSMs implement straightforward and exact solutions to problems that transformers struggle to represent exactly. They can also model bounded hierarchical structure with optimal memory even without simulating a stack. On the other hand, we identify a design choice in current SSMs that limits their expressive power. We discuss implications for SSM and LM research, and verify results empirically on a recent SSM, Mamba.

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Summary

  • The paper demonstrates that SSMs effectively model formal languages, successfully tracking Flip Flop state transitions with optimal memory efficiency.
  • The paper establishes that SSMs and transformers cover distinct regions of the TC circuit complexity class, highlighting differences in handling regular languages like PARITY.
  • The paper theoretically characterizes non-time-invariant SSMs as modeling star-free regular languages, offering a clear criterion for addressing finite-state problems.

Overview of the Expressive Capacity of State Space Models: A Formal Language Perspective

This paper undertakes a detailed examination of the expressive capacity of State Space Models (SSMs) in the context of LLMing, comparing them to transformers and traditional recurrent neural networks (RNNs). While transformers have rapidly risen to prominence due to their parallelized training advantages and strong empirical performance, SSMs have emerged as a competitive alternative, potentially offering capabilities that transformers inherently lack. The paper applies a formal language theoretical lens to uncover the underlying in-principle abilities of SSMs, providing insights into their strengths and limitations relative to other architectures.

Key Contributions

  1. Expressive Capacity in Formal Languages:
    • The paper explores the ability of SSMs to model different classes of formal languages, effectively framing the discussion in terms of language classes traditionally used to understand computational problems.
    • A core finding is that while SSMs and transformers cover overlapping yet distinct fragments of the TC circuit complexity class, there are specific areas where each architecture struggles or excels.
  2. Differences Between SSMs and Transformers:
    • It is demonstrated that SSMs can handle certain regular languages and bounded hierarchical structures with optimal memory efficiency. An example is the successful modeling of Flip Flop state tracking, where SSMs provide simple and exact solutions, contrasting with the empirical difficulties transformers face in generalization.
    • Conversely, SSMs face limitations with modular counting in regular languages, as observed in their struggle with the PARITY function, a problem that requires more advanced state-tracking capabilities not inherently supported by the current design of SSMs.
  3. Theoretical Characterization of Expressive Power:
    • The paper identifies that the expressive power of non-time-invariant SSMs with nonnegative gates corresponds to the class of star-free regular languages. This finding aligns with the Krohn-Rhodes theorem, revealing that SSMs easily model set-reset automata through cascade products without the Kleene star operation.
    • This characterization provides a decisive criterion for the finite-state problems SSMs can solve, simplifying understanding and predictability compared to transformers, for which length generalization remains problematic.

Implications and Future Directions

  • The results presented imply that SSMs could provide distinct advantages in handling certain language tasks, suggesting avenues for hybrid architectures that integrate the strengths of SSMs and transformers.
  • The theoretical implications underscore the potential need to revisit the parametrization of SSMs, especially regarding the handling of nonlinearities and precision, to overcome expressivity bottlenecks.
  • Practically, these insights can guide the design of more efficient and capable LLMs by highlighting scenarios where SSMs outperform or can complement existing transformer models.

Conclusions

The paper delivers a rigorous account of the expressive capacities of SSMs, yielding clear implications for both theoretical computer science and practical AI model development. By employing formal language frameworks, the research elucidates strengths and weaknesses in SSM design choices and suggests future directions for leveraging the unique characteristics of SSM-based architectures. The findings advocate for explorations into hybrid models combining SSMs with other architectures, potentially unlocking new capabilities in LLMing tasks.

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