Learning under Distributional Drift: Prequential Reproducibility as an Intrinsic Statistical Resource

TL;DR

This paper introduces an intrinsic drift budget to quantify distributional changes in learning environments, providing a rate-based characterization of reproducibility and bounds on the effects of drift on performance.

Contribution

It develops a Fisher-Rao distance-based framework to separate environmental change from feedback effects, establishing tight bounds and invariance results for prequential reproducibility under distributional drift.

Findings

The average Fisher-Rao motion rate $C_T/T$ characterizes the impact of drift on performance.

A drift-feedback bound of order $T^{-1/2}+C_T/T$ is proven, showing the effect of drift on reproducibility.

Experiments demonstrate that monitoring channels can retain drift signals even when the data law is unknown.

Abstract

Statistical learning under distributional drift remains poorly characterized, especially in closed-loop settings where learning alters the data-generating law. We introduce an intrinsic drift budget $C_T$ that quantifies cumulative information-geometric motion of the data distribution along the realized learner-environment trajectory, measured in Fisher-Rao distance. The budget separates exogenous environmental change from policy-sensitive feedback induced by the learner's actions. This gives a rate-based characterization of prequential reproducibility: when performance on the realized stream is used to predict one-step-ahead performance under the next distribution, the drift contribution enters through the average motion rate $C_T/T$, not through cumulative drift alone. We prove a drift-feedback bound of order $T^{-1/2}+C_T/T$, up to controlled second-order remainder terms, and establish a matching sharpness lower bound for the same prequential reproducibility gap on a canonical regular subclass. Thus the dependence on the average Fisher-Rao motion rate is tight up to constants: $C_T/T$ is sufficient for upper control and unavoidable on regular hard subclasses. We further prove an information-theoretic indistinguishability result showing that order-$C/T$ effects on the one-step-ahead target need not be identifiable from the realized performance stream alone. Finally, we show that fixed monitoring channels induce contracted observable Fisher motion, and experiments, including a misspecified real-data feedback setting, indicate that appropriately chosen channels can retain risk-relevant drift signal when the intrinsic data-generating law is unavailable. The resulting theory treats exogenous drift, adaptive data analysis, and performative feedback as different sources of Fisher-Rao motion along the same learner-environment trajectory.