How Smooth Is Attention? Article Swipe

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Valérie Castin , Pierre Ablin , Gabriel Peyré · YOU? · · 2023 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2312.14820

Self-attention and masked self-attention are at the heart of Transformers' outstanding success. Still, our mathematical understanding of attention, in particular of its Lipschitz properties - which are key when it comes to analyzing robustness and expressive power - is incomplete. We provide a detailed study of the Lipschitz constant of self-attention in several practical scenarios, discussing the impact of the sequence length $n$ and layer normalization on the local Lipschitz constant of both unmasked and masked self-attention. In particular, we show that for inputs of length $n$ in any compact set, the Lipschitz constant of self-attention is bounded by $\sqrt{n}$ up to a constant factor and that this bound is tight for reasonable sequence lengths. When the sequence length $n$ is too large for the previous bound to be tight, which we refer to as the mean-field regime, we provide an upper bound and a matching lower bound which are independent of $n$. Our mean-field framework for masked self-attention is novel and of independent interest. Our experiments on pretrained and randomly initialized BERT and GPT-2 support our theoretical findings.

Type

preprint

Language

en

Landing Page

http://arxiv.org/abs/2312.14820

PDF

https://arxiv.org/pdf/2312.14820

OA Status

green

Related Works

10

OpenAlex ID

https://openalex.org/W4390215368

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OpenAlex ID

https://openalex.org/W4390215368

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DOI

https://doi.org/10.48550/arxiv.2312.14820

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Title

How Smooth Is Attention?

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Type

preprint

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Language

en

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Publication year

2023

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Publication date

2023-12-22

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Authors

Valérie Castin, Pierre Ablin, Gabriel Peyré

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Landing page

https://arxiv.org/abs/2312.14820

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https://arxiv.org/pdf/2312.14820

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Open access

Yes

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OA status

green

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OA URL

https://arxiv.org/pdf/2312.14820

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Concepts

Lipschitz continuity, Robustness (evolution), Upper and lower bounds, Computer science, Probability measure, Mathematics, Measure (data warehouse), Constant (computer programming), Artificial neural network, Mathematical optimization, Artificial intelligence, Discrete mathematics, Mathematical analysis, Gene, Biochemistry, Programming language, Database, Chemistry

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0

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Related works (count)

10

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