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Theoretical Foundations of Deep Selective State-Space Models (2402.19047v4)

Published 29 Feb 2024 in cs.LG and math.DS

Abstract: Structured state-space models (SSMs) such as S4, stemming from the seminal work of Gu et al., are gaining popularity as effective approaches for modeling sequential data. Deep SSMs demonstrate outstanding performance across a diverse set of domains, at a reduced training and inference cost compared to attention-based transformers. Recent developments show that if the linear recurrence powering SSMs allows for multiplicative interactions between inputs and hidden states (e.g. GateLoop, Mamba, GLA), then the resulting architecture can surpass in both in accuracy and efficiency attention-powered foundation models trained on text, at scales of billion parameters. In this paper, we give theoretical grounding to this recent finding using tools from Rough Path Theory: we show that when random linear recurrences are equipped with simple input-controlled transitions (selectivity mechanism), then the hidden state is provably a low-dimensional projection of a powerful mathematical object called the signature of the input -- capturing non-linear interactions between tokens at distinct timescales. Our theory not only motivates the success of modern selective state-space models such as Mamba but also provides a solid framework to understand the expressive power of future SSM variants.

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Citations (13)
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Summary

  • The paper introduces selective state-space models with input-controlled transitions that capture complex sequential patterns using Rough Path Theory.
  • The paper proves that wide, randomly initialized models in this framework are fully expressive, approximating any continuous function of the input sequence.
  • The paper offers practical insights on model stability and computational efficiency, recommending the use of diagonal recurrences for optimized performance.

Theoretical Analysis and Expressivity of Selective State-Space Models in Sequential Data Processing

Introduction

State-Space Models (SSMs), notably recognized for their simplified structure and performance in sequential data analysis, have recently undergone significant advancements. The emergence of models such as Mamba and its selective variants represents the cutting-edge in leveraging SSMs for effective integration within deep learning architectures. This paper delineates a comprehensive theoretical foundation for understanding the expressivity and implementation of such models. Through an exploration of Rough Path Theory, the paper elucidates how linear recurrences equipped with input-controlled transitions can encapsulate complex data patterns in lower-dimensional projections, ultimately enhancing their predictive capabilities.

Foundations of Selective SSMs

Selective State-Space Models (S-SSMs) augment traditional SSMs by incorporating input-dependent transitions, which facilitate selective attention to relevant portions of the input sequence. This mechanism is integral to models such as Mamba and its contemporaries. By dynamically adjusting the model's linear transition based on the input, S-SSMs demonstrate an impressive ability to emphasize salient features while suppressing extraneous information. This capability significantly boosts performance across various domains, including LLMing and time-series forecasting.

Expressivity through Rough Path Theory

Leveraging Rough Path Theory, the paper provides a rigorous framework for analyzing the expressivity of S-SSMs. The key takeaway is that the hidden state in these models encodes a rich set of input statistics captured through the signature of the input path. This finding underlines the theoretical foundation for the effectiveness of input-controlled transitions in S-SSMs. Furthermore, the theoretical analysis reveals that wide, randomly initialized models are fully expressive, capable of approximating any continuous function of the input sequence.

Implications and Practical Considerations

The theoretical insights gained shed light on the design and optimization of future SSM variants. The framework suggests that the expressivity of S-SSMs is primarily attributed to their ability to capture nonlinear interactions between distinct timescales of the input sequence. Notably, the paper also discusses stability considerations, crucial for training deep SSMs, and suggests practical guidelines for model implementation, including the relevance of diagonal recurrences for computational efficiency.

Conclusions and Future Directions

This research elucidates the substantial expressive power of SSMs, especially when augmented with input-controlled transitions. These findings not only justify the recent success of selective SSMs but also pave the way for future explorations into enhancing their capabilities. The incorporation of Rough Path Theory into the analysis of SSMs opens novel avenues for theoretical investigations, promising further advancements in the field of sequential data processing.

Analytical Perspectives on Expressivity

Substantial emphasis is placed on a rigorous approach to understanding expressivity within S-SSMs. Through meticulous analysis, the paper delineates the conditions under which these models can encapsulate input relationships of varying complexity. This aspect of the research underscores the potential for S-SSMs to revolutionize the processing of sequential data by offering a theoretically grounded approach to capturing the inherent dynamics of diverse datasets.

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