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

Sofiya Zaichyk

arxiv: 2512.13506 · v4 · pith:KBTEC6NJnew · submitted 2025-12-15 · 💻 cs.LG · stat.ML

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

Sofiya Zaichyk This is my paper

Pith reviewed 2026-05-21 16:54 UTC · model grok-4.3

classification 💻 cs.LG stat.ML

keywords distributional driftprequential reproducibilityFisher-Rao distancedrift budgetstatistical learningclosed-loop learningperformative feedbackinformation-geometric motion

0 comments

The pith

Prequential reproducibility gaps under distributional drift are bounded by T^{-1/2} plus the average Fisher-Rao motion rate C_T/T.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces an intrinsic drift budget C_T that measures the total Fisher-Rao distance the data distribution travels along the trajectory created by a learner interacting with its environment. This budget distinguishes changes that come from outside the system from changes caused by the learner's own actions. Using this measure the authors derive that the gap between performance on observed data and predicted performance on the next distribution depends on the average rate of motion C_T/T rather than the total accumulated drift. They prove an upper bound of order T^{-1/2} + C_T/T with controlled remainders and show a matching lower bound on a canonical regular subclass of distributions, establishing that the average rate term is both sufficient and necessary up to constants.

Core claim

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-0

What carries the argument

The intrinsic drift budget C_T, which quantifies the cumulative Fisher-Rao motion of the data distribution along the learner-environment trajectory and separates exogenous drift from learner-induced feedback.

If this is right

The drift contribution to the reproducibility gap enters through the average motion rate C_T/T rather than total cumulative drift.
Order-C/T effects on the one-step-ahead target need not be identifiable from the realized performance stream alone.
Fixed monitoring channels induce contracted observable Fisher motion while appropriately chosen channels can retain risk-relevant drift signal.
The theory treats exogenous drift, adaptive data analysis, and performative feedback uniformly as sources of Fisher-Rao motion on the same trajectory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same geometric accounting of motion might extend to other online performance metrics such as regret or calibration error.
In misspecified real-data settings, channel selection that preserves drift signal could improve one-step-ahead risk estimates without recovering the full data-generating law.
The indistinguishability result implies that certain feedback effects remain undetectable from performance observations alone, suggesting the need for auxiliary monitoring.

Load-bearing premise

The lower bound establishing tightness of the C_T/T term holds specifically on a canonical regular subclass of distributions.

What would settle it

A concrete example on a regular distribution subclass in which the prequential reproducibility gap scales differently from the predicted T^{-1/2} + C_T/T order would falsify the claimed tightness.

Figures

Figures reproduced from arXiv: 2512.13506 by Sofiya Zaichyk.

**Figure 1.** Figure 1: The learner’s policy πt and exogenous influence ηt act on the environment, evolving θt to θt+1 under F(θt , ut , ηt). This new state defines the next data distribution pθt+1 , closing the feedback loop. Exogenous factors ηt perturb θt externally, while endogenous feedback arises from the learner’s own actions. estimated becomes path-dependent. In such settings, the limiting factor is not only sample size, … view at source ↗

**Figure 2.** Figure 2: Geometric intuition for learning under drift. (a) At each step the environment parameter [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Linear–Gaussian demonstrations for the decomposition [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗

**Figure 4.** Figure 4: Neural-network teacher–learner validation. [PITH_FULL_IMAGE:figures/full_fig_p027_4.png] view at source ↗

**Figure 5.** Figure 5: Observed ∆ rep T versus predictions from the plane fit trained on T ̸= Thold (with Thold = 6400 highlighted). where B ∈ R k×d is a fixed random projection (row-normalized) and σK > 0 is channel noise. The induced output law Qt := K#Pt remains Gaussian, Qt = N (Bθt , BΣB⊤ + σ 2 KIk), and therefore its Fisher–Rao step length is again closed form: dF (Qt+1, Qt) = ∥B(θt+1 − θt)∥(BΣB⊤+σ 2 KIk)−1 , A (K) T = X T… view at source ↗

**Figure 6.** Figure 6: Observable Fisher motion under a fixed monitoring channel [PITH_FULL_IMAGE:figures/full_fig_p029_6.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines an intrinsic drift budget C_T in Fisher-Rao distance along the learner trajectory and derives a prequential reproducibility bound of T^{-1/2} + C_T/T with a matching lower bound on a regular subclass.

read the letter

The central new element is the intrinsic drift budget C_T, which measures cumulative Fisher-Rao motion along the actual realized path and separates exogenous change from learner-induced feedback. This leads to an upper bound on the prequential gap of order T^{-1/2} plus C_T/T, plus a matching lower bound on a canonical regular subclass and an information-theoretic result that order-C/T effects need not be identifiable from the observed stream alone. The framework also notes that fixed monitoring channels contract the observable motion. That unification of exogenous drift, adaptive analysis, and performative effects under one geometric quantity is the clearest contribution. The abstract states the bounds come with controlled second-order remainders, which is a reasonable level of care for this style of result. The lower bound is explicitly restricted to the regular subclass, so the claim that C_T/T is unavoidable holds only inside that class. If the subclass excludes vanishing Fisher information, non-smooth paths, or high-dimensional regimes that matter in practice, then the tightness statement does not extend to the broader settings where distributional drift is usually discussed. Without the full derivations it is hard to judge how restrictive the regularity conditions really are or whether the remainder controls are uniform. The work is aimed at researchers who already think in information geometry or who need rate-based guarantees for closed-loop systems. A reader looking for a fresh way to quantify drift in online or performative settings would find the construction useful even if the lower-bound scope needs tightening. I would send it to referees; the specific rate and the geometric unification are concrete enough to merit review, though the authors should clarify how far the sharpness result travels outside the regular subclass.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces an intrinsic drift budget C_T quantifying cumulative Fisher-Rao motion along the realized learner-environment trajectory in closed-loop statistical learning under distributional drift. It separates exogenous environmental change from policy-induced feedback and derives a prequential reproducibility bound of order T^{-1/2} + C_T/T (with controlled second-order remainders). A matching lower bound is shown on a canonical regular subclass, an information-theoretic indistinguishability result is proved, and fixed monitoring channels are shown to contract observable Fisher motion; experiments on a misspecified real-data feedback setting are included.

Significance. If the derivations hold, the work supplies a geometrically grounded rate characterization that unifies exogenous drift, adaptive data analysis, and performative effects as different sources of Fisher-Rao motion. The explicit dependence on the average motion rate C_T/T, the tightness result on the regular subclass, and the indistinguishability theorem are substantive contributions. The experimental validation on real data with monitoring channels adds practical value. The intrinsic, trajectory-based definition of C_T avoids post-hoc fitting and is a clear strength.

major comments (2)

[Abstract and lower-bound theorem] Abstract and lower-bound section: the claim that C_T/T is 'unavoidable' rests on the matching lower bound holding on a 'canonical regular subclass.' The manuscript should clarify whether this subclass is representative of standard parametric families with positive-definite Fisher information and smooth drift paths, or whether the tightness may fail outside it (e.g., vanishing information, non-smooth trajectories, or high-dimensional regimes). This directly affects the scope of the central tightness assertion.
[Upper-bound theorem] Upper-bound derivation: the stated rate T^{-1/2} + C_T/T holds 'up to controlled second-order remainder terms.' Explicit bounds or conditions ensuring these remainders are o(T^{-1/2} + C_T/T) uniformly along the trajectory should be stated, as they are load-bearing for the claimed order.

minor comments (2)

[Definition of C_T] Notation for C_T: the separation of exogenous versus policy-sensitive components should be made fully explicit in the definition, with a short remark on how the realized trajectory is used to compute the budget.
[Experimental results] Experiments: the misspecified real-data setting is useful; adding quantitative comparison of risk-relevant drift signal retained by different monitoring channels would strengthen the section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback on our manuscript. We address each major comment below and plan to incorporate clarifications and explicit statements in the revised version.

read point-by-point responses

Referee: [Abstract and lower-bound theorem] Abstract and lower-bound section: the claim that C_T/T is 'unavoidable' rests on the matching lower bound holding on a 'canonical regular subclass.' The manuscript should clarify whether this subclass is representative of standard parametric families with positive-definite Fisher information and smooth drift paths, or whether the tightness may fail outside it (e.g., vanishing information, non-smooth trajectories, or high-dimensional regimes). This directly affects the scope of the central tightness assertion.

Authors: We agree that the scope of the lower-bound result requires clarification to avoid overgeneralization. The 'canonical regular subclass' is intended to capture standard parametric families equipped with a positive-definite Fisher information matrix and smooth (e.g., Lipschitz) drift paths in the parameter space. The matching lower bound relies on these regularity conditions to ensure the information-geometric distance behaves appropriately. We will revise the abstract and the lower-bound theorem statement to explicitly list these assumptions and note that the tightness may not hold in regimes with vanishing Fisher information, non-smooth trajectories, or certain high-dimensional settings where additional logarithmic factors could appear. This revision will better delineate the applicability of the unavoidability claim. revision: yes
Referee: [Upper-bound theorem] Upper-bound derivation: the stated rate T^{-1/2} + C_T/T holds 'up to controlled second-order remainder terms.' Explicit bounds or conditions ensuring these remainders are o(T^{-1/2} + C_T/T) uniformly along the trajectory should be stated, as they are load-bearing for the claimed order.

Authors: The upper-bound derivation in the manuscript controls the second-order remainder terms via standard regularity assumptions from information geometry, specifically bounded third-order derivatives of the log-likelihood function along the trajectory and a uniform bound on the curvature of the statistical manifold. Under these conditions, the remainders are indeed o(T^{-1/2} + C_T/T) uniformly. We will update the theorem statement to include these explicit conditions and provide the precise order of the remainder term in the revised manuscript, ensuring the claimed rate is rigorously justified without hidden assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation of prequential bounds

full rationale

The paper defines the intrinsic drift budget C_T directly as the cumulative Fisher-Rao distance along the realized learner-environment trajectory, separating exogenous change from policy feedback. The upper bound of order T^{-1/2} + C_T/T is derived in terms of this externally measured geometric quantity, and the matching lower bound is shown on a specified canonical regular subclass. No step reduces the target reproducibility gap to a fitted parameter or self-referential definition by construction; C_T is not obtained by fitting to the gap itself. No load-bearing self-citations, uniqueness theorems imported from prior author work, or ansatzes smuggled via citation are present in the abstract or described derivation chain. The theory treats drift, adaptivity, and feedback as sources of motion along a common trajectory using standard information-geometric tools, making the claims self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the newly introduced drift budget C_T and standard background results from information geometry and statistical learning theory; no free parameters are fitted to data and no additional invented entities beyond C_T are postulated.

axioms (2)

standard math Fisher-Rao distance defines a valid Riemannian metric on the space of probability distributions
Invoked to measure cumulative motion along the learner-environment trajectory.
domain assumption Existence of a canonical regular subclass of distributions on which the lower bound holds
Required for the matching sharpness result stated in the abstract.

invented entities (1)

intrinsic drift budget C_T no independent evidence
purpose: Quantifies cumulative information-geometric motion of the data distribution along the realized learner-environment trajectory and separates exogenous from feedback-induced change
Newly defined quantity that serves as the central object in the drift-feedback bounds.

pith-pipeline@v0.9.0 · 5819 in / 1733 out tokens · 61958 ms · 2026-05-21T16:54:25.469562+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Fisher-Rao is the unique Riemannian metric on parametric families that is invariant under smooth reparameterizations (Čencov, 1982).

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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Raef Bassily, Kobbi Nissim, Adam Smith, Thomas Steinke, Uri Stemmer, and Jonathan Ullman

ISBN 978-0-8218-4302-4. Raef Bassily, Kobbi Nissim, Adam Smith, Thomas Steinke, Uri Stemmer, and Jonathan Ullman. Algorithmic stability for adaptive data analysis? In Yishay Mansour and Daniel Wichs, editors, Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2016), pages 1046–1059, Cambridge, MA, USA,

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doi: 10.1145/2897518.2897566

Association for Computing Machinery. doi: 10.1145/2897518.2897566. Omar Besbes, Yonatan Gur, and Assaf Zeevi. Non-stationary stochastic optimization.Operations Research, 63(5):1227–1244,

work page doi:10.1145/2897518.2897566

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Gilles Blanchard, Aniket Anand Deshmukh, Urun Doğan, Gyemin Lee, and Clayton Scott

doi: 10.1287/opre.2015.1408. Gilles Blanchard, Aniket Anand Deshmukh, Urun Doğan, Gyemin Lee, and Clayton Scott. Domain generalization by marginal transfer learning.Journal of Machine Learning Research, 22(17-679): 1–55,

work page doi:10.1287/opre.2015.1408 2015

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Eric Hall and Rebecca Willett

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work page doi:10.1103/physrevlett.118.010601

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Jiachun Li, David Simchi-Levi, and Yunxiao Zhao

doi: 10.1007/b98852. Jiachun Li, David Simchi-Levi, and Yunxiao Zhao. Optimal adaptive experimental design for estimating treatment effect,

work page doi:10.1007/b98852

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ISBN 9783642341052

Springer-Verlag. ISBN 9783642341052. doi: 10.1007/978-3-642-34106-9_13. Noboru Murata, Motoaki Kawanabe, Andreas Ziehe, Klaus-Robert Müller, and Shun ichi Amari. On- line learning in changing environments with applications in supervised and unsupervised learning. Neural Networks, 15(4):743–760,

work page doi:10.1007/978-3-642-34106-9_13

[7] [7]

doi: 10.1016/S0893-6080(02)00060-6

ISSN 0893-6080. doi: 10.1016/S0893-6080(02)00060-6. Pedro A. Ortega and Daniel A. Braun. Thermodynamics as a theory of decision-making with information-processing costs.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 469(2153),

work page doi:10.1016/s0893-6080(02)00060-6

[8] [8]

doi: 10.1098/rspa.2012.0683. Juan C. Perdomo, Tijana Zrnic, Celestine Mendler-Dünner, and Moritz Hardt. Performative prediction. InProceedings of the 37th International Conference on Machine Learning (ICML), pages 7599–7609, Online,

work page doi:10.1098/rspa.2012.0683 2012

[9] [9]

Richard S

doi: 10.1103/PhysRevLett.109.120604. Richard S. Sutton and Andrew G. Barto.Reinforcement Learning: An Introduction. The MIT Press, Cambridge, MA, second edition,

work page doi:10.1103/physrevlett.109.120604

[10] [10]

Peng Zhao, Yu-Jie Zhang, Lijun Zhang, and Zhi-Hua Zhou

doi: 10.1007/978-1-4612-1880-7_29. Peng Zhao, Yu-Jie Zhang, Lijun Zhang, and Zhi-Hua Zhou. Adaptivity and non-stationarity: Problem-dependent dynamic regret for online convex optimization.Journal of Machine Learning Research, 25(98):1–52,

work page doi:10.1007/978-1-4612-1880-7_29