Adaptive Mean Estimation by In-Context Learning: A Gradient-Flow Analysis

Prior Fitted Networks (PFNs) such as TabPFN zhang2025tabpfnonemodel now rival established statistical procedures across prediction and estimation tasks. A natural explanation is that PFNs have the property of statistical adaptivity, that is they perform nearly as well as a method tailored to the true data-generating model for a heterogeneous set of models, while not being told which model the data comes from. Prior work does not give a conclusive explanation, how gradient-based pretraining learns an adaptive estimator. We study how such adaptivity is learned in a controlled location-estimation problem. Each task is an unlabeled sample whose family is hidden: Gaussian data call for averaging, with error of order $n^{-1}$, whereas uniform data are best estimated from their extremes, at the faster rate $n^{-2}$. We also provide the example of a symmetric Gaussian mixture, for which a rate of ${\sigma }_n^2/n$ can be attained. On scalar inputs, softmax attention computes the derivative of the empirical cumulant-generating function. A single primitive therefore both supplies features that distinguish the families and forms estimators interpolating between the sample mean and the mid-range. Building on this, we combine attention experts through either a softmax mixture of experts or a gated linear unit (GLU), and analyze stagewise gradient flow. With $\tilde{\Omega }(n^{1+ϵ})$ pretraining tasks, the learned estimator is asymptotically efficient on Gaussian tasks, within a factor $n^{ϵ}$ of the minimax rate on uniform tasks, and order-optimal on mixtures in a shrinking-variance regime. These guarantees extend to new locations and longer contexts. A risk decomposition separates expert error, routing error and normalization error, which clarifies the architectural contrast. Softmax gating enforces normalization and exact translation equivariance, whereas the GLU must learn it: its dynamics separate into fast bias removal followed by slow expert selection. End-to-end experiments recover the predicted specialization.