Theoretical Bounds for Stable In-Context Learning

Abstract: In-context learning (ICL) is flexible but its reliability is highly sensitive to prompt length. This paper establishes a non-asymptotic lower bound that links the minimal number of demonstrations to ICL stability under fixed high-dimensional sub-Gaussian representations. The bound gives explicit sufficient conditions in terms of spectral properties of the covariance, providing a computable criterion for practice. Building on this analysis, we propose a two-stage observable estimator with a one-shot calibration that produces practitioner-ready prompt-length estimates without distributional priors. Experiments across diverse datasets, encoders, and generators show close alignment between the predicted thresholds and empirical knee-points, with the theory acting as a conservative but reliable upper bound; the calibrated variant further tightens this gap. These results connect spectral coverage to stable ICL, bridge theory and deployment, and improve the interpretability and reliability of large-scale prompting in realistic finite-sample regimes.