Exact Sequence Interpolation with Transformers
Albert Alcalde, Giovanni Fantuzzi, Enrique Zuazua
TL;DR
This work proves that transformers can exactly interpolate real-valued sequence-to-sequence mappings with outputs of length $m^j$, independent of the input lengths, via an explicit construction that alternates feed-forward and self-attention layers. It establishes both hardmax and softmax self-attention regimes, achieving exact interpolation with block counts $L = 2\sum m^j + 2N + 1$ (hardmax) or $L = 2\sum m^j + 3N$ (softmax) and parameter counts $P = O(d\sum m^j)$, using low-rank or identity-like attention matrices. The methodology hinges on a sequencing strategy: separate overlapping sequences, select leaders to represent outputs, collapse tokens through clustering, and interpolate onto the target sequences, with precise constructions for both tight mathematical control and practical relevance. These results illuminate why transformers perform well on long-input, short-output tasks and offer insights into regularized training dynamics, including linear scaling of optimality with the regularization parameter when exact interpolation is achievable.
Abstract
We prove that transformers can exactly interpolate datasets of finite input sequences in $\mathbb{R}^d$, $d\geq 2$, with corresponding output sequences of smaller or equal length. Specifically, given $N$ sequences of arbitrary but finite lengths in $\mathbb{R}^d$ and output sequences of lengths $m^1, \dots, m^N \in \mathbb{N}$, we construct a transformer with $\mathcal{O}(\sum_{j=1}^N m^j)$ blocks and $\mathcal{O}(d \sum_{j=1}^N m^j)$ parameters that exactly interpolates the dataset. Our construction provides complexity estimates that are independent of the input sequence length, by alternating feed-forward and self-attention layers and by capitalizing on the clustering effect inherent to the latter. Our novel constructive method also uses low-rank parameter matrices in the self-attention mechanism, a common feature of practical transformer implementations. These results are first established in the hardmax self-attention setting, where the geometric structure permits an explicit and quantitative analysis, and are then extended to the softmax setting. Finally, we demonstrate the applicability of our exact interpolation construction to learning problems, in particular by providing convergence guarantees to a global minimizer under regularized training strategies. Our analysis contributes to the theoretical understanding of transformer models, offering an explanation for their excellent performance in exact sequence-to-sequence interpolation tasks.