On the Expressiveness and Length Generalization of Selective State-Space Models on Regular Languages (2412.19350v1)

Abstract: Selective state-space models (SSMs) are an emerging alternative to the Transformer, offering the unique advantage of parallel training and sequential inference. Although these models have shown promising performance on a variety of tasks, their formal expressiveness and length generalization properties remain underexplored. In this work, we provide insight into the workings of selective SSMs by analyzing their expressiveness and length generalization performance on regular language tasks, i.e., finite-state automaton (FSA) emulation. We address certain limitations of modern SSM-based architectures by introducing the Selective Dense State-Space Model (SD-SSM), the first selective SSM that exhibits perfect length generalization on a set of various regular language tasks using a single layer. It utilizes a dictionary of dense transition matrices, a softmax selection mechanism that creates a convex combination of dictionary matrices at each time step, and a readout consisting of layer normalization followed by a linear map. We then proceed to evaluate variants of diagonal selective SSMs by considering their empirical performance on commutative and non-commutative automata. We explain the experimental results with theoretical considerations. Our code is available at https://github.com/IBM/selective-dense-state-space-model.

• The paper introduces SD-SSM, leveraging dense transition matrices to achieve near-perfect length generalization on regular language tasks.
• It compares SD-SSM with RNNs, LSTMs, and Transformers, highlighting significant improvements in modeling finite-state automata dynamics.
• The study offers theoretical insights into the limitations of diagonal selective SSMs and underscores the benefits of parallel training for scalable sequence modeling.

An Analysis of Expressiveness and Generalization in Selective State-Space Models

The paper "On the Expressiveness and Length Generalization of Selective State-Space Models on Regular Languages" investigates the efficacy of Selective State-Space Models (SSMs) in capturing the computational complexities of regular languages and generalizing across varied input lengths. With the predominant position of Transformers in natural language processing, the paper pivots towards alternatives like SSMs that offer parallel training with sequential inference capabilities. The work not only assesses current limitations in modern SSM architectures but also introduces the Selective Dense State-Space Model (SD-SSM) as an innovative framework capable of achieving exemplary length generalization.

The research delineates a nuanced landscape where existing SSMs have been less than adept at accurately modeling and generalizing finite-state automata (FSA) dynamics. While Transformers have been the architecture of choice for language processing tasks, their inability to track state changes in FSAs is a well-documented limitation in theoretical studies. This paper posits that though nonlinear recurrent neural networks (RNNs) can overcome such hurdles, their lack of parallelizability limits computational feasibility for long sequences.

Key Contributions

1. Introduction of the SD-SSM: The paper unveils SD-SSM, a model leveraging dense transition matrices created via a softmax aggregation of dictionary matrices at each time step. This approach is complemented by a readout process involving layer normalization and linear mapping, which significantly bolsters the model's length generalization capabilities. Achieving near-perfect results on various regular language tasks using a single layer, SD-SSM demonstrates superior length generalization where other models falter.
2. Comparison and Evaluation: The empirical paper contrasts models like RNN, LSTM, S4D, H3, Mamba, and others, discerning their performance on different automata tasks. SD-SSM’s ability to generalize across longer input sequences, beyond those encountered during training, underscores its robustness compared to the lackluster results observed in Transformers and other SMMs.
3. Theoretical Insights into Diagonal Selective SSMs: The exploration extends to selective SSMs with complex-diagonal transition matrices, highlighting their performance disparities on automata defined by commutative versus non-commutative operations. Such differentiation in empirical performance aligns with theoretical characterizations, suggesting an inherent limitation in expressiveness due to the reliance on diagonal matrices.
4. Computational Efficiency: SD-SSM's architecture benefits from the parallel scan algorithm for efficient state computations, reinforcing the model’s scalability in parallel processing scenarios as opposed to traditional sequential evaluations. Time efficiency comparisons reveal marked improvements over conventional recurrent computations, asserting SD-SSM’s viability for deployment in diverse sequence modeling tasks.

Implications and Future Work

The emergence of SD-SSM presents new opportunities within the field of machine learning, particularly in applications requiring reliable computation of regular languages and complex state transitions. The ability to achieve concurrent top-tier performance with single-layer architectures paves the way for more streamlined, resource-efficient models in AI applications.

Future explorations could further integrate SD-SSM with real-world data scenarios, expanding its applicability beyond theoretical tasks. Enhancements may involve integrating temperature scaling or annealing strategies in the model’s selection mechanism to improve interpretability and application-specific performance. Moreover, bridging the gap between the demonstrated theoretical expressiveness and practical use in varied data domains remains a fertile area for continued research.

In conclusion, this paper stands out for establishing a compelling case for selective SSMs, especially with SD-SSM, as salient alternatives to Transformers. By synthesizing theoretical insights with robust empirical validation, it signals a shift towards more efficient, scalable sequence models that retain expressiveness for intricate language and formal computation tasks.