Numerical Investigation of Sequence Modeling Theory using Controllable Memory Functions (2506.05678v2)

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 memory function and a parameter that determines the strength of temporal dependence. This setup allows us to produce a continuum of tasks that vary in temporal complexity, enabling fine-grained analysis of model behavior concerning specific memory properties. We focus on four representative memory functions, each corresponding to a distinct class of temporal structures. Experiments on several sequence modeling architectures confirm existing theoretical insights and reveal new findings. 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.

• The paper introduces a synthetic benchmarking framework that evaluates sequence models on diverse temporal dependencies using controlled memory functions such as exponential decay and polynomial decay.
• The study finds that recurrent models like LSTMs and S4Ds excel with rapidly decaying dependencies yet struggle with complex patterns like polynomial decay, exposing architectural limitations.
• The evaluation reveals that convolutional networks and transformers require careful tuning, especially in attention mechanisms, to efficiently approximate variable temporal structures.

Insights on Sequence Modeling Evaluation Using Controllable Memory Functions

Sequence modeling is an essential facet of machine learning, relevant across multiple domains such as NLP, time-series analysis, and dynamic system modeling. This paper, authored by Jiang, Bao, Wang, and Li, aims to address the complexities involved in modeling diverse temporal dependencies implicit in sequential data by proposing a novel synthetic benchmarking framework. Their motivation stems from a gap in understanding the mathematical behaviors and limitations of popular sequence model architectures such as RNNs, Transformers, and SSMs.

Synthetic Benchmarking Framework

The authors introduce a synthetic benchmarking framework designed to evaluate sequence models against distinct temporal structures, characterized by well-defined memory functions with tunable parameters for complexity. They exploit controllable synthetic targets with specific memory properties, allowing for detailed model behavior analysis concerning these properties. They define four representative memory functions: exponential decay, polynomial decay, impulse response, and Airy, each reflecting unique temporal dependencies that simulate different real-world scenarios in data processing tasks.

Key Observations and Experimental Results

The authors systematically evaluate prominent sequence modeling architectures: LSTM, S4D, TCN, and Transformers. Here are some observations highlighted from their empirical work:

1. Recurrent Architectures' Temporal Dependency Handling: The research finds that architectural bias plays a significant role in a model's ability to capture temporal structures. For instance, LSTMs and S4Ds exhibit consistent performance on rapidly decaying temporal dependencies modeled by exponential functions, supporting existing theoretical insights. However, their efficacy diminishes significantly for polynomial decay functions, reiterating known limitations.
2. Convolutional Networks' Behavior: TCNs effectively achieve stable performance across impulse functions due to their dilated convolution structures, capturing long-range dependencies efficiently. Yet, they demonstrate increased approximation errors over Airy functions, suggesting that sparsity in temporal dependencies substantially impacts their efficacy.
3. Effects on Transformer Architectures: Transformers exhibit similar qualitative behaviors to convolutional networks regarding approximation difficulty with varying $\alpha$ values for exponential, polynomial, and Airy functions. The authors link this to insights around the attention matrix rank, proposing that rank influences approximation efficacy significantly.
4. Trade-offs in Multi-Head Attention: Notably, a non-linear relationship between the number of attention heads and hidden dimension in transformers emerged from experiments, suggesting complex dependencies of temporal structures on model configuration. This relationship indicates a deeper exploration of attention mechanisms in relation to sequence modeling could yield meaningful insights.

Implications and Future Directions

This paper underscores the practical need for synthetic benchmarks that reflect controlled memory characteristics in evaluating sequence models. It provides evidence for theoretical assertions regarding sequence architecture behaviors under diverse temporal structures. Importantly, it highlights directions for further research, including refining complexity measures for convolutional architectures and dissecting the impact of position encodings in transformers.

In future work, expanding this methodology to multi-layer models or integrating interaction effects between multiple temporal filters could add depth to understanding complex data behaviors. Furthermore, this framework could serve as a basis for exploring new language and dynamic system modeling approaches, particularly by extending theoretical models to account for nonlinear real-world structures.

Overall, the authors' contribution in offering a controlled evaluation framework helps bridge the gap between theoretical model behavior research and practical application, paving the way for potentially more efficient and effective sequence modeling architectures.