Longhorn: State Space Models are Amortized Online Learners (2407.14207v5)

Abstract: Modern LLMs are built on sequence modeling via next-token prediction. While the Transformer remains the dominant architecture for sequence modeling, its quadratic decoding complexity in sequence length poses a major limitation. State-space models (SSMs) present a competitive alternative, offering linear decoding efficiency while maintaining parallelism during training. However, most existing SSMs rely on linear recurrence designs that appear somewhat ad hoc. In this work, we explore SSM design through the lens of online learning, conceptualizing SSMs as meta-modules for specific online learning problems. This approach links SSM design to formulating precise online learning objectives, with state transition rules derived from solving these objectives. Based on this insight, we introduce a novel deep SSM architecture, Longhorn, whose update resembles the closed-form solution for solving the online associative recall problem. Our experimental results show that Longhorn outperforms state-of-the-art SSMs, including the Mamba model, on standard sequence modeling benchmarks, LLMing, and vision tasks. Specifically, Longhorn achieves a 1.8x improvement in sample efficiency compared to Mamba, and can extrapolate over contexts that are up to 16x longer during inference.

Longhorn: State Space Models as Amortized Online Learners

The paper "Longhorn: State Space Models are Amortized Online Learners" authored by Bo Liu, Rui Wang, Lemeng Wu, Yihao Feng, Peter Stone, and Qiang Liu, explores the core challenges and advancements in sequence modeling for AI, particularly focusing on alternatives to the Transformer architecture. The authors propose a novel framework positioning state-space models (SSMs) through the lens of online learning. This approach facilitates a conceptualization of SSMs as meta-modules aimed at optimizing specific online learning objectives.

Abstract and Introduction Overview

The motivation behind this research stems from the computational inefficiencies inherent in Transformers, primarily their quadratic growth in computational cost with respect to sequence length. Despite improvements in aspects like efficient decoding and memory optimization, scaling Transformers for lengthy context windows remains problematic. The paper suggests that SSMs could offer more efficient alternatives for sequence modeling due to their linear decoding efficiency and high parallelizability. However, a guiding principle for SSM design has been lacking.

Proposed Approach and Contributions

The paper makes significant contributions by proposing a theoretical framework that conceptualizes SSMs as solving online learning problems. This perspective shifts the design focus towards creating online learning objectives, thereby deriving state transition rules from these objectives. Based on this principle, the paper introduces Longhorn, a deep SSM architecture derived from the implicit update for an online regression problem.

The Longhorn Model

Longhorn's architecture is grounded in the objective of online associative recall. This design choice leverages the closed-form solution to an online learning objective, leading to a recurrence relation that inherently embodies stability without needing manually designed gating mechanisms. Specifically, Longhorn's recurrence relation during inference is obtained through:

$$S_t = (1 - \Delta_t \otimes k_t^{\odot 2}) \odot S_{t-1} + (\Delta_t \odot x_t ) \otimes k_t$$

where $\Delta_t$ is the step size determined by the online learning objective. This structure ensures that the model does not require a separately parameterized forget gate, thus saving parameters while maintaining or enhancing performance.

Empirical Results

The empirical results in the paper are particularly compelling. Longhorn outperforms state-of-the-art SSMs, including the Mamba model, across standard sequence modeling benchmarks and LLMing tasks. Notably, Longhorn achieves a 1.8x improvement in sampling efficiency and demonstrates remarkable extrapolation capabilities, being able to handle longer context lengths without significant performance degradation.

Comparative Analysis

The paper also offers a comparative analysis against other SSM variants such as Linear Attention Models (e.g., Gated Linear Attention, Mamba, and Griffin) and Fast Weight Programmers. Each model's recurrence relation is interpreted through the online learning framework, providing a coherent understanding of their design and guiding principles.

Implications and Future Work

The implications of this research extend both practically and theoretically. Practically, the reduction in parameter count and the improved efficiency of Longhorn make it a viable candidate for large-scale sequence modeling tasks. Theoretically, the online learning framework offers a structured approach to SSM design, potentially leading to further innovations in this space.

Moving forward, the paper suggests exploring other online learning objectives that align with modern hardware capabilities. Additionally, integrating sliding-window attention mechanisms, as suggested in recent studies, could further enhance the performance of Longhorn.

Conclusion

In conclusion, "Longhorn: State Space Models are Amortized Online Learners" presents a robust framework and a novel model that addresses key inefficiencies in existing sequence modeling approaches. By framing SSMs through the online learning paradigm, the authors provide a clear, efficient, and theoretically grounded method to improve sequence modeling tasks, setting the stage for further advancements in the field.