Theoretical limitations of multi-layer Transformer (2412.02975v1)

Abstract: Transformers, especially the decoder-only variants, are the backbone of most modern LLMs; yet we do not have much understanding of their expressive power except for the simple $1$-layer case. Due to the difficulty of analyzing multi-layer models, all previous work relies on unproven complexity conjectures to show limitations for multi-layer Transformers. In this work, we prove the first $\textit{unconditional}$ lower bound against multi-layer decoder-only transformers. For any constant $L$, we prove that any $L$-layer decoder-only transformer needs a polynomial model dimension ($n^{\Omega(1)}$) to perform sequential composition of $L$ functions over an input of $n$ tokens. As a consequence, our results give: (1) the first depth-width trade-off for multi-layer transformers, exhibiting that the $L$-step composition task is exponentially harder for $L$-layer models compared to $(L+1)$-layer ones; (2) an unconditional separation between encoder and decoder, exhibiting a hard task for decoders that can be solved by an exponentially shallower and smaller encoder; (3) a provable advantage of chain-of-thought, exhibiting a task that becomes exponentially easier with chain-of-thought. On the technical side, we propose the multi-party $\textit{autoregressive}$ $\textit{communication}$ $\textit{model}$ that captures the computation of a decoder-only Transformer. We also introduce a new proof technique that finds a certain $\textit{indistinguishable}$ $\textit{decomposition}$ of all possible inputs iteratively for proving lower bounds in this model. We believe our new communication model and proof technique will be helpful to further understand the computational power of transformers.

• The paper presents the first unconditional lower bounds for multi-layer Transformer architectures using a novel multi-party autoregressive communication model.
• It demonstrates a depth-width trade-off by showing that an L-layer model requires significantly more parameters than an (L+1)-layer model for certain compositional tasks.
• The study reveals that chain-of-thought methods can exponentially simplify tasks, providing the first provable advantage for Transformer decoders.

Theoretical Limitations of Multi-layer Transformers: A Critical Evaluation

In this paper, the authors undertake a substantial investigation into the theoretical limitations of multi-layer Transformer architectures. Building on previous work that has primarily focused on single-layer Transformers, they extend the analysis to multi-layer models, an area that has posed significant challenges due to the absence of proven complexity theoretical bounds. Traditionally reliant on speculative complexity conjectures, the approach developed in this research provides the first unconditional lower bounds for multi-layer Transformer architectures.

Main Theoretical Insights

The core contribution of this work is in identifying and proving a polynomial model dimension requirement for any L-layer decoder-only Transformer attempting to execute the sequential composition of L functions over an input consisting of n tokens. Notably, the authors introduce and make use of the multi-party autoregressive communication model to capture the computation processes inherent to a decoder-only Transformer. Furthermore, the introduction of a novel proof technique allows for the identification of an indistinguishable decomposition of all potential inputs, which is pivotal in establishing lower bounds within this model.

Technical and Computational Findings

The paper asserts three primary results:

1. Depth-Width Trade-off: The first depth-width trade-off is established for multi-layer Transformers, illustrating that an L-layer model necessitates significantly more parameters than an (L + 1)-layer model for certain compositional tasks.
2. Encoder-Decoder Separation: The paper presents an unconditional separation between encoder and decoder architectures. A task exists that poses considerable difficulty for decoders and can be solved by an encoder with exponentially fewer layers and parameters.
3. Chain-of-Thought (CoT) Advantage: The findings suggest a task that becomes exponentially simpler when approached using a chain-of-thought methodology, providing the first provable benefits of CoT for Transformers.

Implications and Speculative Future Directions

The implications of these findings are multifaceted:

• Theoretical Advances: By establishing new lower bounds without relying on conjectures, this work paves the way for understanding the intrinsic computational capabilities and limitations of multi-layer Transformers. This advance mitigates some of the theoretical obstacles previously conjectured, such as circuit lower bound barriers.
• Architectural Design: The insights regarding depth-width trade-offs could influence the future design of Transformer architectures, particularly emphasizing the potential computational advantages of deeper models over wider ones.
• Chain-of-Thought Utilization: The recognition of CoT benefits could steer the development of tasks benefiting from step-by-step reasoning methodologies, impacting both practical applications and the theoretical exploration of Transformer models.

Conclusion

The paper makes seminal contributions by establishing concrete limitations for multi-layer Transformers, thus advancing the formal understanding of these architectures in computational complexity. By presenting significant trade-offs and leveraging novel theoretical tools, it opens new avenues for both theoretical examination and practical application in future AI system designs. The authors' methodologies and findings form a crucial foundation for ongoing research and understanding of the deep learning constructs that underpin many modern AI technologies.