Efficiently Modeling Long Sequences with Structured State Spaces (2111.00396v3)

Abstract: A central goal of sequence modeling is designing a single principled model that can address sequence data across a range of modalities and tasks, particularly on long-range dependencies. Although conventional models including RNNs, CNNs, and Transformers have specialized variants for capturing long dependencies, they still struggle to scale to very long sequences of $10000$ or more steps. A promising recent approach proposed modeling sequences by simulating the fundamental state space model (SSM) ( x'(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t) ), and showed that for appropriate choices of the state matrix ( A ), this system could handle long-range dependencies mathematically and empirically. However, this method has prohibitive computation and memory requirements, rendering it infeasible as a general sequence modeling solution. We propose the Structured State Space sequence model (S4) based on a new parameterization for the SSM, and show that it can be computed much more efficiently than prior approaches while preserving their theoretical strengths. Our technique involves conditioning ( A ) with a low-rank correction, allowing it to be diagonalized stably and reducing the SSM to the well-studied computation of a Cauchy kernel. S4 achieves strong empirical results across a diverse range of established benchmarks, including (i) 91\% accuracy on sequential CIFAR-10 with no data augmentation or auxiliary losses, on par with a larger 2-D ResNet, (ii) substantially closing the gap to Transformers on image and LLMing tasks, while performing generation $60\times$ faster (iii) SoTA on every task from the Long Range Arena benchmark, including solving the challenging Path-X task of length 16k that all prior work fails on, while being as efficient as all competitors.

Efficiently Modeling Long Sequences with Structured State Spaces

The paper "Efficiently Modeling Long Sequences with Structured State Spaces" by Albert Gu, Karan Goel, and Christopher Ré introduces the Structured State Space (S4) model, addressing the challenge of sequence modeling, particularly in capturing long-range dependencies (LRDs). Existing models like RNNs, CNNs, and Transformers have limitations in scaling efficiently to very long sequences (e.g., over 10,000 time steps). Despite many optimizations, they still struggle with memory and computation efficiency when faced with extended sequence lengths.

Abstract

The authors propose the S4 model, which builds upon state space models (SSMs). These are mathematical frameworks traditionally used across various scientific disciplines, described by the state equation $x'(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t)$. Prior work demonstrated that appropriate choices of the state matrix $A$ could effectively handle LRDs. However, their significant computational and memory demands hinder practical use. S4 introduces a novel parameterization that makes SSMs computationally viable by conditioning the state matrix $A$ with a low-rank correction, allowing its stable diagonalization. This reduces the SSM to the efficient computation of a Cauchy kernel, significantly improving performance.

Impact and Results

S4 achieves strong empirical results across various benchmarks:

1. CIFAR-10: Reaches 91% accuracy in a sequential setup without data augmentation, comparable to a larger 2-D ResNet.
2. Transformer Tasks: Closes performance gaps on image and LLMing tasks, while being 60 times faster in generation.
3. Long Range Arena (LRA): Sets new state-of-the-art (SoTA) on all tasks, including the challenging Path-X task of length 16k, where all previous models failed.

These results highlight S4's capacity to handle LRDs efficiently, making it a practical model for diverse sequence modeling tasks.

Methodology

S4's innovative approach involves a structured reparameterization of the state space model:

1. State Space Models (SSMs): Utilizes the theory of continuous-time memorization through HiPPO matrices, which manage the history of inputs by specific choices of the state matrix $A$.
2. Parameterization: Reconstitutes $A$ as a sum of a normal matrix and a low-rank correction. The low-rank component is adjusted using Woodbury’s identity, while the normal component is diagonalized stably.
3. Computation Efficiency: Reducing the problem to computing a Cauchy kernel allows leveraging efficient algorithms, resulting in computation complexity of $\tilde{O}(N+L)$ and memory usage of $O(N+L)$.

Empirical Validation

1. Long Range Arena: S4 demonstrates superior performance on LRA, comprehensively outperforming previous models on all six tasks, averaging 86.09% accuracy. Notably, it solves the Path-X task, showcasing its ability to manage very long sequences.
2. Speech Classification: Successfully classifies raw speech data (length 16,000) with 98.32% accuracy, outperforming even sophisticated models with extensive preprocessing.

Implications

• General-purpose Sequence Model: S4 shows potential as a versatile sequence model, simplifying the traditionally high specialization required for tasks in different domains (e.g., text, images, audio). Its ability to merge effectively continuous-time, convolutions, and recurrence-based models can unify capabilities into a single model framework.
• Future Work: Anticipated developments include enhancing S4's efficiency further, exploring larger-scale applications, and extending the model to accommodate higher-dimensional data for more complex tasks such as video modeling.

Conclusion

The authors address a significant bottleneck in sequence modeling by making SSMs computationally viable for long sequences through innovative parameterization and efficient algorithmic techniques. S4 stands out both in theory and practice, significantly advancing the capability to model long-range dependencies efficiently.

In summarizing, S4 represents a robust advancement in sequence modeling, with broad applicability across diverse domains and tasks, setting a new standard for handling long-range dependencies.