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

arXiv: https://arxiv.org/abs/2512.13506 · HTML: https://mlsanity.com/paper/2512.13506/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 42 references that are in the map.
- First posted on arXiv 15 Dec 2025.

## Builds on (most cited first, 12 of 12)

- [Performative Prediction](https://mlsanity.com/paper/2002.06673/) (2020)
- [Information-theoretic analysis of generalization capability of learning algorithms](https://mlsanity.com/paper/1705.07809/) (2017)
- [Domain Generalization by Marginal Transfer Learning](https://mlsanity.com/paper/1711.07910/) (2017)
- [Non-stationary Stochastic Optimization](https://mlsanity.com/paper/1307.5449/) (2013)
- [Algorithmic Stability for Adaptive Data Analysis](https://mlsanity.com/paper/1511.02513/) (2015)
- [Generalization in Adaptive Data Analysis and Holdout Reuse](https://mlsanity.com/paper/1506.02629/) (2015)
- [How to Learn when Data Reacts to Your Model: Performative Gradient Descent](https://mlsanity.com/paper/2102.07698/) (2021)
- [Adaptivity and Non-stationarity: Problem-dependent Dynamic Regret for Online Convex Optimization](https://mlsanity.com/paper/2112.14368/) (2021)
- [Dynamical Models and Tracking Regret in Online Convex Programming](https://mlsanity.com/paper/1301.1254/) (2013)
- [Fisher Information and Natural Gradient Learning of Random Deep Networks](https://mlsanity.com/paper/1808.07172/) (2018)
- [Guaranteed Validity for Empirical Approaches to Adaptive Data Analysis](https://mlsanity.com/paper/1906.09231/) (2019)
- [Optimal Adaptive Experimental Design for Estimating Treatment Effect](https://mlsanity.com/paper/2410.05552/) (2024)

## Abstract (arXiv)

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 CT 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 CT/T, not through cumulative drift alone. We prove a drift-feedback bound of order T-1/2+CT/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: CT/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.