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

We investigate the statistical and computational limits of prompt tuning for transformer-based foundation models. Our key contributions are prompt tuning on \textit{single-head} transformers with only a \textit{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 \textit{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.

Outlier-Efficient Hopfield Layers for Large Transformer-Based Models

We introduce an Outlier-Efficient Modern Hopfield Model (termed $\mathtt{OutEffHop}$) and use it to address the outlier-induced challenge of quantizing gigantic transformer-based models. Our main contribution is a novel associative memory model facilitating \textit{outlier-efficient} associative memory retrievals. Interestingly, this memory model manifests a model-based interpretation of an outlier-efficient attention mechanism ($\text{Softmax}_1$): it is an approximation of the memory retrieval process of $\mathtt{OutEffHop}$. Methodologically, this allows us to debut novel outlier-efficient Hopfield layers a powerful attention alternative with superior post-quantization performance. Theoretically, the Outlier-Efficient Modern Hopfield Model retains and improves the desirable properties of the standard modern Hopfield models, including fixed point convergence and exponential storage capacity. Empirically, we demonstrate the proposed model's efficacy across large-scale transformer-based and Hopfield-based models (including BERT, OPT, ViT and STanHop-Net), benchmarking against state-of-the-art methods including $\mathtt{Clipped\_Softmax}$ and $\mathtt{Gated\_Attention}$. Notably, $\mathtt{OutEffHop}$ achieves on average $\sim$22+\% reductions in both average kurtosis and maximum infinity norm of model outputs accross 4 models.