Asymptotic Study of In-context Learning with Random Transformers through Equivalent Models

We study the in-context learning (ICL) capabilities of pretrained Transformers in the setting of nonlinear regression. Specifically, we focus on a random Transformer with a nonlinear MLP head where the first layer is randomly initialized and fixed while the second layer is trained. Furthermore, we consider an asymptotic regime where the context length, input dimension, hidden dimension, number of training tasks, and number of training samples jointly grow. In this setting, we show that the random Transformer behaves equivalent to a finite-degree Hermite polynomial model in terms of ICL error. This equivalence is validated through simulations across varying activation functions, context lengths, hidden layer widths (revealing a double-descent phenomenon), and regularization settings. Our results offer theoretical and empirical insights into when and how MLP layers enhance ICL, and how nonlinearity and over-parameterization influence model performance.