Transformers Can Learn Posterior Predictive Distributions In-Context

Mechanistic accounts of in-context learning (ICL) have identified iterative algorithms for linear regression and related linear prediction tasks, often using linear or ReLU attention variants. For nonlinear ICL, prior work has related softmax and kernelized attention to functional-gradient-type dynamics, but it remains unclear whether a standard transformer with softmax attention can implement a convergent solver with an end-to-end prediction-error guarantee. In this paper, we study in-context kernel ridge regression (KRR) with Gaussian kernels and show that a standard softmax-attention transformer can approximate the KRR predictor during its forward pass by implementing preconditioned Richardson iteration on the associated kernel linear system. Under bounded-data assumptions, we construct a single-head transformer with $O(\log(1/ε))$ blocks and MLP width $O(\sqrt{N/ε})$ that achieves $ε$-accurate prediction for prompts of length $N$. Our construction reveals a functional decomposition within the transformer architecture: softmax attention produces a row-normalized Gaussian-kernel operator needed for cross-token interactions, while ReLU MLP layers act locally to approximate the intra-token scalar arithmetic required by the update. Empirically, we train GPT-2-style transformers on Gaussian-process regression tasks to further test the preconditioned Richardson interpretation. Through linear probing, we compare the transformer's layer-wise predictions with the step-wise outputs of classical KRR solvers and find that its error profiles align most consistently with preconditioned Richardson iteration. Ablation studies further support this interpretation. Together, our theory and experiments identify preconditioned Richardson iteration as a concrete mechanism that softmax-attention transformers can realize for nonlinear in-context Gaussian-kernel regression.

Transformers have demonstrated remarkable in-context learning (ICL) capabilities. The strong ICL performance of transformers is commonly believed to arise from their ability to implicitly execute certain algorithms on the context, thereby enhancing prediction and generation. In this work, we investigate how transformers with softmax attention perform in-context learning on linear classification data. We first construct a class of multi-layer transformers that can perform in-context logistic regression, with each layer exactly performing one step of normalized gradient descent on an in-context loss. Then, we show that our constructed transformer can be obtained through (i) training a single self-attention layer supervised by one-step gradient descent, and (ii) recurrently applying the trained layer to obtain a looped model. Training convergence guarantees of the self-attention layer and out-of-distribution generalization guarantees of the looped model are provided. Our results advance the theoretical understanding of ICL mechanism by showcasing how softmax transformers can effectively act as in-context learners.

Bayesian predictive inference provides a principled framework for uncertainty quantification, data efficiency, and robust generalization. However, exact inference is often intractable, and scalable approximations may remain computationally expensive or require restrictive modeling assumptions that degrade predictive performance. Prior-Data Fitted and in-context models have recently emerged as an amortized alternative by learning to map datasets directly to predictive distributions, but existing approaches are tightly coupled to the support of the training prior and lack explicit mechanisms for adapting to new priors at test time, resulting in limited robustness under distribution shift. We introduce a multi-task in-context learning framework for amortized hierarchical Bayesian predictive inference that explicitly represents prior information as a prefix of in-context datasets. A transformer trained on sequences of prior and target tasks learns to adapt its predictions across families of priors. On a suite of evaluations with increasing difficulty, including out-of-meta-distribution priors and priors with high-dimensional latent structures, our method matches oracle Bayesian predictors while being orders of magnitude faster. We further demonstrate its practical relevance on a real-world spatiotemporal temperature prediction benchmark. Code is available at https://github.com/martianmartina/multi-task-bayesian-icl/.