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

arXiv:2512.13506v3 Announce Type: replace-cross
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 the cumulative information-geometric motion of the data distribution along the realized learner-environment trajectory, measured in Fisher-Rao distance (the Riemannian metric induced by Fisher information on a statistical manifold of data-generating laws). The budget decomposes this motion into exogenous change (environmental drift that would occur without intervention) and policy-sensitive feedback contributions (drift induced by the learner’s actions through the closed loop). This yields a rate-based characterization: in prequential reproducibility bounds — where performance on the realized stream is used to predict one-step-ahead performance under the next distribution — the drift contribution enters through the average drift rate $C_T/T$, i.e., normalized cumulative Fisher-Rao motion per time step. We prove a drift–feedback bound of order $T^{-1/2} + C_T/T$ (up to a controlled second-order remainder) and establish a matching minimax lower bound on a canonical subclass, showing this dependence is tight up to constants. Consequently, when $C_T/T$ is nonnegligible, one-step-ahead reproducibility admits an irreducible accuracy floor of the same order. Finally, the framework places exogenous drift, adaptive data analysis, and performative feedback within a common geometric account of distributional motion.