Transformers Can Represent $n$-gram Language Models

Exploring Probabilistic Representational Capacities of Transformer Language Models in Relation to n-gram Language Models

Introduction

Transformer models, particularly for language tasks, have exhibited significant capabilities and versatility. However, many aspects of their theoretical foundations, especially their representational capacities in modeling probabilistic distributions over strings, remain less explored. The discussed study aims to bridge this gap by establishing a concrete relationship between transformer language models (LMs) and n-gram LMs—a well-known class of probabilistic language models. The core thesis of this investigation centers around demonstrating that transformer LMs with either hard or sparse attention can exactly represent any n-gram LM, providing a significant insight into their lower bounds in terms of probabilistic representational capacity.

Representation Analysis

Attention Mechanisms and n-gram Implementation

The study elucidates how transformers leveraging hard and sparse attention mechanisms can be configured to represent n-gram LMs. For hard attention, transformers with a number of heads or layers equal to n-1 can simulate an n-gram model effectively. This setup either allocates each head to focus on a specific position in the input sequence or uses sequential layering to capture positional information incrementally across layers.

In contrast, sparse attention employs a differentiable approach that approximates the hard attention's selection process, ensuring that each head still focuses predominantly on a single preceding symbol position. This model requires unbounded positional encodings and non-linear transformations for accurate performance, diverging from standard settings but maintaining a close analog to hard attention mechanisms.

Encoding and Complexity

The paper provides a rigorous analysis of how transformers encode the necessary information from the past n-1 symbols to compute the probability of the subsequent symbol, adherent to the n-gram assumption. This involves a detailed look into the size and complexity of contextual representations and highlights the model's reliance on extensive one-hot encodings to mimic n-gram behavior effectively.

Theoretical Contributions and Implications

Probabilistic Capacity and Transformers

By confirming that transformer LMs can indeed simulate basic n-gram LMs under certain configurations, the paper establishes a foundational understanding of the minimum capabilities of transformer models in probabilistic language processing. This result not only enriches the theoretical landscape of neural networks but also sparks further inquiry into the more nuanced and complex probabilistic models transformers might accommodate.

Practical Modeling Considerations

While the theoretical framework presented uses assumptions like hard attention and idealized encoding methods, which are not prevalent in practical applications, the insights garnered provide a valuable perspective on what foundational probabilistic tasks transformers are inherently capable of when abstracted from application-specific optimizations and restrictions.

Future Directions in AI Research

Moving forward, this revelation prompts additional questions about the upper bounds of transformer capabilities and how these models manage more sophisticated probabilistic distributions beyond n-gram limits. Moreover, understanding the learning dynamics of such theoretically possible representations from real-world data and their implications on model training and performance becomes an essential next step.

Conclusion

The analysis adds a significant piece to the puzzle of understanding transformer models by linking them to a classical model of computation—the n-gram model. It forms a basis for evaluating transformers not just as practical tools but as subjects of theoretical study in the broader AI research community, exploring the depths of their computational and representational capabilities in formal terms.