Learning to Adapt: In-Context Learning Beyond Stationarity

Pre-trained transformers are able to learn from examples provided as part of the prompt without any weight updates, a remarkable ability known as in-context learning (ICL). Despite its demonstrated efficacy across various domains, the theoretical understanding of ICL is still developing. Whereas most existing theory has focused on linear models, we study ICL in the nonlinear regression setting. Through the interaction mechanism in attention, we explicitly construct transformer networks to realize nonlinear features, such as polynomial or spline bases, which span a wide class of functions. Based on this construction, we establish a framework to analyze end-to-end in-context nonlinear regression with the constructed features. Our theory provides finite-sample generalization error bounds in terms of context length and training set size. We numerically validate the theory on synthetic regression tasks.

Recent work has demonstrated that transformers and linear attention models can perform in-context learning (ICL) on simple function classes, such as linear regression. In this paper, we empirically study how these two attention mechanisms differ in their ICL behavior on the canonical linear-regression task of Garg et al. We evaluate learning quality (MSE), convergence, and generalization behavior of each architecture. We also analyze how increasing model depth affects ICL performance. Our results illustrate both the similarities and limitations of linear attention relative to quadratic attention in this setting.

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 and constructive implementation of the oracle kernel estimators for these function classes using a wide and shallow transformer architecture, illustrating how the prediction is carried out by the interaction between the query token and the context tokens. Our work complements existing studies by providing numerical and theoretical evidence of how the attention mechanism adapts to anisotropic kernels in both linear and nonlinear settings.