Transformers are Deep Infinite-Dimensional Non-Mercer Binary Kernel Machines
Matthew A. Wright, Joseph E. Gonzalez
TL;DR
The paper reframes Transformer dot-product attention as an indefinite, asymmetric kernel operating between two input domains within a pair of reproducing kernel Banach spaces, revealing an infinite-dimensional feature space. It introduces a binary RKBS learning framework with a representer theorem and a universal approximation result, showing that Transformers can approximate any binary RKBS mapping. An approximate kernel-learning approach via distillation is proposed to scale the method, connecting attention logits to kernel feature maps learned by neural nets. Empirical results indicate that infinite-dimensional kernels (e.g., exponentiated dot-product, RBF) can improve performance on machine translation, suggesting kernel choice is a meaningful design knob for Transformer architectures.
Abstract
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.