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

Martin Eppert; Krishna Balasubramanian; Subhro Ghosh; Jason Klusowski; Yan Shuo Tan

cs.LGOct 6, 2026

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

Authors: Martin Eppert, Krishna Balasubramanian, Subhro Ghosh, Jason Klusowski, Yan Shuo Tan

Organizations: National University of Singapore · University of California, Davis · Princeton University

Abstract

Prior Fitted Networks (PFNs) such as TabPFN 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. 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 $σ^2_n/n$ 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. We combine attention experts through either a softmax mixture of experts or a gated linear unit (GLU), and analyze stagewise gradient flow. With $\widetildeΩ(n^{1+ε})$ 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.

Figures & tables

Panel of Figure 1 — Figure 1: Illustration of our studied architecture. We also highlight the parts which were fixed for simplicity.

Alternative family	terminal $\|\lambda_{2}\|$	Gaussian risk	Alternative risk	$R_{\mathrm{A},n}^{\star}$
Uniform	$n^{1-\epsilon}$	$(1+o(1))/n$	$\mathcal{O}(n^{-2+\epsilon})$	$\asymp n^{-2}$
Gaussian mixture	$\frac{\log(1/\sigma_{n}^{2})}{4\sqrt{1-\sigma_{n}^{2}}}$	$(1+o(1))/n$	$\mathcal{O}(\sigma_{n}^{2}/n)$	$\geq\sigma_{n}^{2}/n$

Table 1: The terminal expert parameters and fresh-context risks certified by the main theorem.

Figure 2: Softmax MoE, joint training with the uniform alternative. Left: $(\bar{w}_{1\mathrm{G}},\bar{w}_{1\mathrm{A}})$ across runs, colored by iteration. Middle: median $|\lambda_{1}|$ and $|\lambda_{2}|$ , where thin lines are individual runs. Right: median validation error of $\widehat{\mu}$ on Gaussian and uniform contexts, with the sample mean and sample mid-range as references. Shading spans all ten runs.

Figure 3: GLU, joint training with the uniform alternative, with panels and conventions as in Figure 2 .

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4: Softmax MoE, joint training with the Gaussian-mixture alternative, as in Figure 2 . The dashed line in the middle panel marks $\log(1/\sigma_{n}^{2})/(4\sqrt{1-\sigma_{n}^{2}})$ .

Figure 5: GLU, joint training with the Gaussian-mixture alternative, as in Figure 3 . The dashed line in the middle panel marks $\log(1/\sigma_{n}^{2})/(4\sqrt{1-\sigma_{n}^{2}})$ .

Figure 6: GLU, joint training with the uniform alternative: population gates

\bar{w}_{kj}

(left) and normalization error

|\bar{w}_{1j}+\bar{w}_{2j}-1|

(right). Curves show medians, and shading spans all ten runs.

Figure 7: GLU, joint training with the Gaussian-mixture alternative, with panels and conventions as in Figure 6 .

Figure 8: First stage with the uniform alternative: $(\bar{w}_{1j},\bar{w}_{2j})$ for Gaussian (left) and uniform (right) contexts across runs, colored by iteration. The dashed line is the critical manifold $\bar{w}_{1j}+\bar{w}_{2j}=1$ .

Figure 9: Uniform alternative: validation error of each expert

g_{k}

and of

\widehat{\mu}

on Gaussian (left) and uniform (right) contexts, with the sample mean and sample mid-range as references. The vertical line marks

t_{\theta}

Figure 10: Uniform alternative: validation error of each expert

g_{k}

and of

\widehat{\mu}

on Gaussian (left) and uniform (right) contexts, repeated for comparison with Figure 12 . The vertical line marks

t_{\theta}

Figure 11: First stage with the Gaussian-mixture alternative: $(\bar{w}_{1j},\bar{w}_{2j})$ for Gaussian (left) and Gaussian-mixture (right) contexts across runs, colored by iteration. The dashed line is the critical manifold $\bar{w}_{1j}+\bar{w}_{2j}=1$ .

Figure 12: Gaussian-mixture alternative: validation error of each expert

g_{k}

and of

\widehat{\mu}

on Gaussian (left) and Gaussian-mixture (right) contexts. The vertical line marks

t_{\theta}

Figure 13: Less constrained architecture, end-to-end training with the uniform alternative. Left: training loss. Middle and right: validation error on Gaussian and uniform contexts, with the sample mean and sample mid-range as references.

Figure 14: Stream plots of the reduced flow studied in Lemma D.8 . Left: population features with LayerNorm. Right: uncentered population features at

\mu=0

without LayerNorm.

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