Learning under Distributional Drift: Reproducibility as an Intrinsic Statistical Resource

• URL: http://arxiv.org/abs/2512.13506v1
• Date: Mon, 15 Dec 2025 16:34:47 GMT
• Title: Learning under Distributional Drift: Reproducibility as an Intrinsic Statistical Resource
• Authors: Sofiya Zaichyk,
• Abstract summary: We introduce a new statistical primitive, the budget $C_T$, which quantifies a system's finite capacity for statistical analysis.<n>No algorithm can achieve smaller worst-case generalization error than that imposed by the average Fisher-Rao drift rate.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $C_T$, 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, $C_T$ 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 $O(T^{-1/2} + C_T/T)$, 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 $C_T/T$ of the data-generating process. The framework situates exogenous drift, adaptive data analysis, and performative prediction within a common geometric structure, with $C_T$ emerging as the intrinsic quantity measuring distributional motion across these settings.

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