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

Keywords: Foundation Model, Prompt Tuning, Transformer, Universal Approximation, Memory Capacity, Computational Efficiency, Fine-Grained Complexity

TL;DR: We study the statistical and computational limits of prompt tuning in single-layer, single-head transformer, showing it is universal seq2seq approximator and, supports almost linear efficient inference.

Abstract: We investigate the statistical and computational limits of prompt tuning for transformer-based foundation models. Our key contributions are that prompt tuning on *single-head* transformers with only a *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 the 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 *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.

Primary Area: foundation or frontier models, including LLMs

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Submission Number: 4923