Paper
Learning under Distributional Drift: Reproducibility as an Intrinsic Statistical Resource
Authors
Sofiya Zaichyk
Abstract
Statistical learning under distributional drift remains insufficiently characterized: when each observation alters the data-generating law, classical generalization bounds can collapse. We introduce a new statistical primitive, the reproducibility budget , which quantifies a system's finite capacity for statistical reproducibility - the extent to which its sampling process can remain governed by a consistent underlying distribution in the presence of both exogenous change and endogenous feedback. Formally, is defined as the cumulative Fisher-Rao path length of the coupled learner-environment evolution, measuring the total distributional motion accumulated during learning. From this construct we derive a drift-feedback generalization bound of order , and we prove a matching minimax lower bound showing that this rate is minimax-optimal. Consequently, the results establish a reproducibility speed limit: no algorithm can achieve smaller worst-case generalization error than that imposed by the average Fisher-Rao drift rate of the data-generating process. The framework situates exogenous drift, adaptive data analysis, and performative prediction within a common geometric structure, with emerging as the intrinsic quantity measuring distributional motion across these settings.