Transformers are Deep Infinite-Dimensional Non-Mercer Binary Kernel Machines
This provides a new theoretical foundation for understanding Transformers, a critical but poorly understood model in AI, though it is incremental in offering a perspective rather than a practical breakthrough.
The paper tackles the problem of understanding Transformers by showing that their dot-product attention can be characterized as a kernel learning method with an infinite-dimensional kernel, and it proves representer and universal approximation theorems for binary non-Mercer kernels, with experimental results suggesting this infinite dimensionality contributes to performance.
Despite their ubiquity in core AI fields like natural language processing, the mechanics of deep attention-based neural networks like the Transformer model are not fully understood. In this article, we present a new perspective towards understanding how Transformers work. In particular, we show that the "dot-product attention" that is the core of the Transformer's operation can be characterized as a kernel learning method on a pair of Banach spaces. In particular, the Transformer's kernel is characterized as having an infinite feature dimension. Along the way we consider an extension of the standard kernel learning problem to a binary setting, where data come from two input domains and a response is defined for every cross-domain pair. We prove a new representer theorem for these binary kernel machines with non-Mercer (indefinite, asymmetric) kernels (implying that the functions learned are elements of reproducing kernel Banach spaces rather than Hilbert spaces), and also prove a new universal approximation theorem showing that the Transformer calculation can learn any binary non-Mercer reproducing kernel Banach space pair. We experiment with new kernels in Transformers, and obtain results that suggest the infinite dimensionality of the standard Transformer kernel is partially responsible for its performance. This paper's results provide a new theoretical understanding of a very important but poorly understood model in modern machine~learning.