Tomoya Wakayama

Tomoya Wakayama, Taiji Suzuki

This paper develops a finite-sample statistical theory for in-context learning (ICL), analyzed within a meta-learning framework that accommodates mixtures of diverse task types. We introduce a principled risk decomposition that separates the total ICL risk into two orthogonal components: Bayes Gap and Posterior Variance. The Bayes Gap quantifies how well the trained model approximates the Bayes-optimal in-context predictor. For a uniform-attention Transformer, we derive a non-asymptotic upper bound on this gap, which explicitly clarifies the dependence on the number of pretraining prompts and their context length. The Posterior Variance is a model-independent risk representing the intrinsic task uncertainty. Our key finding is that this term is determined solely by the difficulty of the true underlying task, while the uncertainty arising from the task mixture vanishes exponentially fast with only a few in-context examples. Together, these results provide a unified view of ICL: the Transformer selects the optimal meta-algorithm during pretraining and rapidly converges to the optimal algorithm for the true task at test time.

Tomoya Wakayama

The remarkable generalization performance of large-scale models has been challenging the conventional wisdom of the statistical learning theory. Although recent theoretical studies have shed light on this behavior in linear models and nonlinear classifiers, a comprehensive understanding of overparameterization in nonlinear regression models is still lacking. This study explores the predictive properties of overparameterized nonlinear regression within the Bayesian framework, extending the methodology of the adaptive prior considering the intrinsic spectral structure of the data. Posterior contraction is established for generalized linear and single-neuron models with Lipschitz continuous activation functions, demonstrating the consistency in the predictions of the proposed approach. Moreover, the Bayesian framework enables uncertainty estimation of the predictions. The proposed method was validated via numerical simulations and a real data application, showing its ability to achieve accurate predictions and reliable uncertainty estimates. This work provides a theoretical understanding of the advantages of overparameterization and a principled Bayesian approach to large nonlinear models.