In-Context Learning of a Linear Transformer Block: Benefits of the MLP Component and One-Step GD Initialization

Ruiqi Zhang, Jingfeng Wu, Peter Bartlett

Keywords: In-Context Learning, Transformers, Approximation Theory, Optimization

TL;DR: We investigate into the in-context learning ability of Linear Transformer Block on linear regression problems and its relationship with one-step gradient descent estimator with learnable initialization.

Abstract:

We study the \emph{in-context learning} (ICL) ability of a \emph{Linear Transformer Block} (LTB) that combines a linear attention component and a linear multi-layer perceptron (MLP) component.
For ICL of linear regression with a Gaussian prior and a \emph{non-zero mean}, we show that LTB can achieve nearly Bayes optimal ICL risk. In contrast, using only linear attention must incur an irreducible additive approximation error.
Furthermore, we establish a correspondence between LTB and one-step gradient descent estimators with learnable initialization ($\mathsf{GD}-\beta$), in the sense that every $\mathsf{GD}-\beta$ estimator can be implemented by an LTB estimator and every optimal LTB estimator that minimizes the in-class ICL risk is effectively a $\mathsf{GD}-\beta$ estimator.
Finally, we show that $\mathsf{GD}-\beta$ estimators can be efficiently optimized with gradient flow, despite a non-convex training objective.
Our results reveal that LTB achieves ICL by implementing $\mathsf{GD}-\beta$, and they highlight the role of MLP layers in reducing approximation error.

Primary Area: Learning theory

Submission Number: 14501