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

Understand

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 ($\mathsf{GD}\text{-}\mathbf{\beta}$), in the sense that every $\mathsf{GD}\text{-}\mathbf{\beta}$ estimator can be implemented by an LTB estimator and every optimal LTB estimator that minimizes the in-class ICL risk is effectively a $\mathsf{GD}\text{-}\mathbf{\beta}$ estimator.