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Computer Science > Machine Learning

arXiv:2412.02975 (cs)
[Submitted on 4 Dec 2024]

Title:Theoretical limitations of multi-layer Transformer

Authors:Lijie Chen, Binghui Peng, Hongxun Wu
View a PDF of the paper titled Theoretical limitations of multi-layer
Transformer, by Lijie Chen and 2 other authors
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    Abstract:Transformers, especially the decoder-only variants, are
    the backbone of most modern large language models; 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.

          Machine Learning (cs.LG); Artificial Intelligence (cs.AI);
Subjects: Computational Complexity (cs.CC); Data Structures and
          Algorithms (cs.DS)
Cite as:  arXiv:2412.02975 [cs.LG]
          (or arXiv:2412.02975v1 [cs.LG] for this version)
          https://doi.org/10.48550/arXiv.2412.02975
          Focus to learn more
          arXiv-issued DOI via DataCite

Submission history

From: Binghui Peng [view email]
[v1] Wed, 4 Dec 2024 02:37:31 UTC (39 KB)
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