Stability of Measure-to-Measure Transformers on Sub-Gaussian Data

Abstract:Transformers have exhibited impressive empirical success across various domains, but their theoretical foundations remain less developed. This work constitutes a mathematical study of the measure-to-measure operators defined by transformers. We show that transformers map sub-Gaussian inputs to sub-Gaussian outputs; this ensures that taking arbitrary-length compositions of the softmax operator is well-defined. We then show that transformers are Hölder continuous with respect to the 1-Wasserstein distance on appropriate spaces of sub-Gaussian inputs. This allows us to establish estimates on the error propagation along a transformer between a sub-Gaussian input and its empirical approximation. We also study a mean-field analog of the cross-attention mechanism, which is an operator from a pair of probability measures to a single probability measure. We show that cross-attention exhibits different Hölder regularity and sample-complexity in its two input arguments. Last, we apply our results to deduce approximation guarantees for measure-to-measure transformers. Together, these results provide a firm stability and finite-sample theory for transformers on sub-Gaussian data.

Machine Learning (stat.ML); Machine Learning (cs.LG)

arXiv:2610.07717 [stat.ML]

(or arXiv:2610.07717v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2610.07717 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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