Transformers as Adaptive Kernel Learners across Tasks for In-Context Learning

Zhaiming Shen, Alexander Cloninger, Rongjie Lai, Wenjing Liao

Transformers have emerged as a dominant architecture in modern machine learning, with in-context learning (ICL) enabling models to adapt to new tasks by conditioning on prompts, without any parameter updates. To understand the attention mechanism in transformer blocks, a growing line of theoretical work interprets ICL through the lens of classical statistical estimators. Recent works have established a connection between the attention mechanism and kernel methods, while the kernel function is usually assumed to be fixed. In practice, the query and key matrices in the attention mechanism are trained, which allows the attention mechanism to serve as an adaptive kernel method. In this work, we demonstrate that transformers can act as adaptive anisotropic kernel learners. Specifically, we study ICL for low-rank linear and multi-index nonlinear regressions, and conduct numerical simulations to show that the learned attention scores closely resemble the oracle kernels for these function classes. We then develop a general framework to analyze the generalization error of ICL by connecting transformers to the oracle kernel estimator for each function class. Our theory gives an explicit…