Stability analysis of transformers on uncertain data distributions

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

Machine Learning

Summary

Transformers are powerful tools used in many technologies, but it is not fully clear why they work so well mathematically. The authors studied how transformers handle data with certain random characteristics called sub-Gaussian inputs. They found that transformers produce outputs that share these characteristics and that their behavior changes smoothly when the input data changes slightly. This research also looked at a special part of transformers called cross-attention and how it processes pairs of data distributions differently. Overall, their work provides mathematical guarantees that transformers behave predictably and reliably when working with this type of uncertain data.

What this means in practice

A theory result. No direct application yet.

Authors

Frank Cole, Nicholas H. Nelsen, Takashi Furuya

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.
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