Numerical Investigation of Sequence Modeling Theory using Controllable Memory Functions (arxiv.org)
arXiv:2506.05678v3 Announce Type: replace
Abstract: The evolution of sequence modeling architectures, from recurrent neural networks and convolutional models to Transformers and structured state-space models, reflects ongoing efforts to address the diverse temporal dependencies inherent in sequential data. Despite this progress, systematically characterizing the strengths and limitations of these architectures remains a fundamental challenge. In this work, we propose a synthetic benchmarking framework to evaluate how effectively different sequence models capture distinct temporal structures. The core of this approach is to generate synthetic targets, each characterized by a parametric memory function $\rho(s, \alpha)$ and a controllable parameter $\alpha$ that determines the temporal strength. This setup allows us to produce a continuum of tasks that vary in temporal complexity, enabling fine-grained analysis of model behavior with respect to specific memory properties. We focus on four representative memory functions, each corresponding to a distinct class of temporal structures: exponential and polynomial functions for decay dynamics, impulse functions for long-range dependencies, and Airy functions for sparsity patterns. Experiments on several sequence modeling architectures confirm existing theoretical insights and reveal new findings regarding approximation capabilities, optimization dynamics, and architectural trade-offs. These results demonstrate the effectiveness of the proposed method in advancing theoretical understanding and highlight the importance of using controllable targets with clearly defined structures for evaluating sequence modeling architectures.
Abstract: The evolution of sequence modeling architectures, from recurrent neural networks and convolutional models to Transformers and structured state-space models, reflects ongoing efforts to address the diverse temporal dependencies inherent in sequential data. Despite this progress, systematically characterizing the strengths and limitations of these architectures remains a fundamental challenge. In this work, we propose a synthetic benchmarking framework to evaluate how effectively different sequence models capture distinct temporal structures. The core of this approach is to generate synthetic targets, each characterized by a parametric memory function $\rho(s, \alpha)$ and a controllable parameter $\alpha$ that determines the temporal strength. This setup allows us to produce a continuum of tasks that vary in temporal complexity, enabling fine-grained analysis of model behavior with respect to specific memory properties. We focus on four representative memory functions, each corresponding to a distinct class of temporal structures: exponential and polynomial functions for decay dynamics, impulse functions for long-range dependencies, and Airy functions for sparsity patterns. Experiments on several sequence modeling architectures confirm existing theoretical insights and reveal new findings regarding approximation capabilities, optimization dynamics, and architectural trade-offs. These results demonstrate the effectiveness of the proposed method in advancing theoretical understanding and highlight the importance of using controllable targets with clearly defined structures for evaluating sequence modeling architectures.
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