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.

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.

W e study the in-context learning (ICL) ability of a Linear Transformer Block (L TB) 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 nonzero mean, we show that L TB 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 L TB and one-step gradient descent estimators with learnable initialization ( GD- β), in the sense that every GD- β estimator can be implemented by an L TB estimator and every optimal L TB estimator that minimizes the in-class ICL risk is effectively a GD- β estimator. Finally, we show that GD- β estimators can be efficiently optimized with gradient flow, despite a non-convex training objective. Our results reveal that L TB achieves ICL by implementing GD- β, and they highlight the role of MLP layers in reducing approximation error.