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arXiv · 2411.16525

Fundamental Limits of Prompt Tuning Transformers: Universality, Capacity and Efficiency

Jerry Yao-Chieh Hu·Wei-Po Wang·Ammar Gilani·Chenyang Li·Zhao Song·Han Liu
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Abstract

We investigate the statistical and computational limits of prompt tuning for transformer-based foundation models. Our key contributions are prompt tuning on \emph{single-head} transformers with only a \emph{single} self-attention layer: (i) is universal, and (ii) supports efficient (even almost-linear time) algorithms under the Strong Exponential Time Hypothesis (SETH). Statistically, we prove that prompt tuning on such simplest possible transformers are universal approximators for sequence-to-sequence Lipschitz functions. In addition, we provide an exponential-in-$dL$ and -in-$(1/\epsilon)$ lower bound on the required soft-prompt tokens for prompt tuning to memorize any dataset with 1-layer, 1-head transformers. Computationally, we identify a phase transition in the efficiency of prompt tuning, determined by the norm of the \emph{soft-prompt-induced} keys and queries, and provide an upper bound criterion. Beyond this criterion, no sub-quadratic (efficient) algorithm for prompt tuning exists under SETH. Within this criterion, we showcase our theory by proving the existence of almost-linear time prompt tuning inference algorithms. These fundamental limits provide important necessary conditions for designing expressive and efficient prompt tuning methods for practitioners.

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Jerry Yao-Chieh Hu, Wei-Po Wang, Ammar Gilani, Chenyang Li, Zhao Song, Han Liu. 2024-11-25. Fundamental Limits of Prompt Tuning Transformers: Universality, Capacity and Efficiency. https://arxiv.org/abs/2411.16525

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