State Space Models are Comparable to Transformers in Estimating Functions with Dynamic Smoothness (2405.19036v1)

Abstract: Deep neural networks based on state space models (SSMs) are attracting much attention in sequence modeling since their computational cost is significantly smaller than that of Transformers. While the capabilities of SSMs have been primarily investigated through experimental comparisons, theoretical understanding of SSMs is still limited. In particular, there is a lack of statistical and quantitative evaluation of whether SSM can replace Transformers. In this paper, we theoretically explore in which tasks SSMs can be alternatives of Transformers from the perspective of estimating sequence-to-sequence functions. We consider the setting where the target function has direction-dependent smoothness and prove that SSMs can estimate such functions with the same convergence rate as Transformers. Additionally, we prove that SSMs can estimate the target function, even if the smoothness changes depending on the input sequence, as well as Transformers. Our results show the possibility that SSMs can replace Transformers when estimating the functions in certain classes that appear in practice.

• The paper demonstrates that structured state space models match transformers in convergence rates when estimating functions with dynamic and piecewise smoothness.
• It rigorously analyzes functions with mixed and anisotropic smoothness, highlighting SSMs’ ability to manage high-dimensional sequence tasks.
• The findings suggest that SSMs offer a computationally efficient alternative for applications such as speech recognition and language processing.

Comparative Analysis of State Space Models and Transformers for Function Estimation with Dynamic Smoothness

The paper "State Space Models are Comparable to Transformers in Estimating Functions with Dynamic Smoothness" by Nishikawa and Suzuki provides a comprehensive theoretical investigation into the potential of Structured State Space Models (SSMs) as alternatives to Transformers for sequence modeling tasks. The paper addresses a critical gap in understanding by focusing on the convergence rates of these models when tasked with estimating sequence-to-sequence functions characterized by dynamic smoothness.

Key Findings and Results

The authors provide robust theoretical evidence indicating that SSMs are capable of estimating functions exhibiting dynamic smoothness with convergence rates comparable to those of Transformers. Specifically, they delve into functions with $\gamma$-smooth and piecewise $\gamma$-smooth structures, illustrating that SSMs can achieve the same convergence rates as Transformers for these classes. This result suggests that, in particular scenarios where the smoothness of a function varies depending on the input sequence, SSMs could indeed be viable substitutes for Transformers.

Two types of smoothness are particularly scrutinized: mixed and anisotropic smoothness. For mixed smoothness, a uniform structure of importance across features is maintained, whereas, for anisotropic smoothness, the level of importance varies across input features. The paper reveals that despite the high dimensionality of inputs and outputs, SSMs maintain their efficacy by leveraging smoothness structures to circumvent the curse of dimensionality—a feature previously well-documented in Transformers.

Implications

This paper has several implications for the field of AI and machine learning, particularly regarding the development of more computationally efficient models for sequence modeling tasks. The significant reduction in computational requirements, due to methods like FFT, enhances the practical applicability of SSMs in contexts with constrained resources. Moreover, this could lead to advancements in domains such as speech recognition and audio generation, where efficient processing of high-dimensional sequential data is paramount.

Furthermore, the paper's exploration into piecewise $\gamma$-smooth functions and the ability of SSMs to adaptively extract features depending on input and output positions broadens the potential use cases for these models. Tasks that require dynamic allocation of attention depending on the context, such as language processing and in-context learning, could benefit from this adaptability.

Conclusions and Future Work

The paper posits that SSMs should be considered as legitimate contenders for certain function estimations, especially where efficiency is paramount and the tasks involve managing changing smoothness across dimensions. However, practical implementation aspects, such as the optimization of such models and their empirical validation across varied datasets, remain open challenges. Future research could focus on refining the efficiency of the parameter tuning processes for these models, potentially expanding their effectiveness and usability across more demanding applications.

In summary, by establishing strong theoretical foundations, this paper opens up avenues for integrating SSMs into more sequence modeling tasks, offering a complementary approach to the well-established Transformer models. This investigation could spark further exploration of efficient alternative architectures in AI, paving the way for more sustainable large-scale data processing solutions.