Exact Sequence Classification with Hardmax Transformers

• URL: http://arxiv.org/abs/2502.02270v1
• Date: Tue, 04 Feb 2025 12:31:00 GMT
• Title: Exact Sequence Classification with Hardmax Transformers
• Authors: Albert Alcalde, Giovanni Fantuzzi, Enrique Zuazua,
• Abstract summary: We prove that hardmax attention transformers perfectly classify datasets of $N$ labeled sequences in $mathbbRd$, $dgeq 2$. Specifically, given $N$ sequences with an arbitrary but finite length in $mathbbRd$, we construct a transformer with $mathcalO(N)$ blocks and $mathcalO(Nd)$ parameters perfectly classifying this dataset.
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• Abstract: We prove that hardmax attention transformers perfectly classify datasets of $N$ labeled sequences in $\mathbb{R}^d$, $d\geq 2$. Specifically, given $N$ sequences with an arbitrary but finite length in $\mathbb{R}^d$, we construct a transformer with $\mathcal{O}(N)$ blocks and $\mathcal{O}(Nd)$ parameters perfectly classifying this dataset. Our construction achieves the best complexity estimate to date, independent of the length of the sequences, by innovatively 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 within the attention mechanism, a common practice in real-life transformer implementations. Consequently, our analysis holds twofold significance: it substantially advances the mathematical theory of transformers and it rigorously justifies their exceptional real-world performance in sequence classification tasks.

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