From Order to Distribution: A Spectral Characterization of Forgetting in Continual Learning

Xingjun Ma; Zonghuan Xu

arxiv: 2604.13460 · v1 · submitted 2026-04-15 · 💻 cs.LG · cs.AI

From Order to Distribution: A Spectral Characterization of Forgetting in Continual Learning

Zonghuan Xu , Xingjun Ma This is my paper

Pith reviewed 2026-05-10 13:55 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords continual learningcatastrophic forgettingspectral characterizationoperator identitylinear regressiontask distributionasymptotic convergence

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The pith

In the linear exact-fit regime with i.i.d. tasks from a distribution, forgetting is exactly characterized by an operator identity with recursive spectral structure.

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

The paper establishes that forgetting in continual learning can be exactly described by an operator identity when tasks are sampled independently from a generating distribution in a linear model. This identity exposes a recursive structure in the spectrum of the task operator, which in turn yields an unconditional upper bound on forgetting, its leading asymptotic behavior, and a precise convergence rate in typical cases. The rate is shown to depend on geometric features of the task distribution itself. A reader would care because this moves the analysis from arbitrary task orders to properties of the underlying distribution, offering a principled way to understand and potentially control forgetting speeds.

Core claim

The central claim is that in an exact-fit linear continual learning setting where tasks are drawn i.i.d. from a task distribution Π, there exists an exact operator identity for the forgetting quantity. This identity reveals a recursive spectral structure. From this, an unconditional upper bound is derived, the leading asymptotic term is identified, and in generic nondegenerate cases the convergence rate is characterized up to constants, with the rate related to geometric properties of Π.

What carries the argument

The exact operator identity for the forgetting quantity, which encodes a recursive spectral structure of the task operator.

If this is right

The forgetting quantity admits an unconditional upper bound that holds regardless of task ordering.
The leading asymptotic term of forgetting is determined by the distribution Π.
The convergence rate of forgetting to its limit is characterized up to constants and depends on the spectral properties of Π.
Geometric properties of the task distribution drive whether forgetting is slow or fast.

Where Pith is reading between the lines

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

This framework suggests that choosing task distributions with favorable geometry could reduce long-term forgetting in sequential learning.
The spectral characterization might inspire similar operator analyses in nonlinear or deep learning models of continual learning.
Estimating the relevant geometric invariants from data could allow prediction of forgetting rates before training.

Load-bearing premise

The derivation assumes tasks are sampled i.i.d. from a fixed distribution and that the linear model exactly fits each task.

What would settle it

Running continual learning on synthetic linear regression tasks sampled i.i.d. from a known distribution Π and verifying whether the observed forgetting matches the predicted upper bound and asymptotic convergence rate.

Figures

Figures reproduced from arXiv: 2604.13460 by Xingjun Ma, Zonghuan Xu.

read the original abstract

A central challenge in continual learning is forgetting, the loss of performance on previously learned tasks induced by sequential adaptation to new ones. While forgetting has been extensively studied empirically, rigorous theoretical characterizations remain limited. A notable step in this direction is \citet{evron2022catastrophic}, which analyzes forgetting under random orderings of a fixed task collection in overparameterized linear regression. We shift the perspective from order to distribution. Rather than asking how a fixed task collection behaves under random orderings, we study an exact-fit linear regime in which tasks are sampled i.i.d.\ from a task distribution~$\Pi$, and ask how the generating distribution itself governs forgetting. In this setting, we derive an exact operator identity for the forgetting quantity, revealing a recursive spectral structure. Building on this identity, we establish an unconditional upper bound, identify the leading asymptotic term, and, in generic nondegenerate cases, characterize the convergence rate up to constants. We further relate this rate to geometric properties of the task distribution, clarifying what drives slow or fast forgetting in this model.

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

This paper derives a new exact operator identity and spectral bounds for forgetting under i.i.d. task sampling in the exact-fit linear regime, but the results stay confined to that model.

read the letter

The main takeaway is that the authors derive an exact operator identity for forgetting in continual learning under i.i.d. sampling from a task distribution in the exact-fit linear regime. This identity exposes a recursive spectral structure and leads to an unconditional upper bound, the leading asymptotic term, and convergence rates in nondegenerate cases. They do a good job shifting the framing from random orderings of fixed tasks to the generating distribution itself. The new identity is not in the cited prior work, and building bounds and rates from it is a direct contribution. Relating the rate to geometric properties of the distribution adds some insight into what controls forgetting speed. The algebra appears solid within the model, with no signs of circularity. They credit the earlier result on orderings and extend it cleanly. The limitation is clear and significant: all results rely on the exact-fit linear i.i.d. model. The operator identity comes from algebraic cancellations in the closed-form linear regression solution. Outside this setting, such as with nonlinear models or non-exact fits, the recursion likely does not hold. The paper does not address robustness or extensions, which keeps the characterization narrow. This is for readers focused on theoretical characterizations of continual learning in simplified linear settings. It offers value to those interested in spectral tools for analyzing forgetting. The work shows honest engagement with the literature and clear derivations, so it deserves a serious referee. I would recommend sending it to peer review, with the expectation that reviewers will probe the assumptions and their implications for broader applicability.

Referee Report

1 major / 2 minor

Summary. The paper analyzes forgetting in continual learning by shifting from random orderings of fixed tasks to the generating distribution Π in an exact-fit linear regression regime with i.i.d. task sampling. It derives an exact operator identity for the forgetting quantity that exposes a recursive spectral structure. From this identity the authors obtain an unconditional upper bound, the leading asymptotic term, and (in generic nondegenerate cases) a characterization of the convergence rate up to constants, relating the rate to geometric properties of Π.

Significance. If the algebraic steps hold, the work supplies a precise spectral tool for analyzing forgetting inside a controlled linear model, clarifying the role of the task distribution's spectrum. The exact identity, unconditional bound, and rate result constitute genuine theoretical contributions that could guide subsequent analysis of distribution-driven forgetting, though the results remain specific to the exact-fit linear i.i.d. setting.

major comments (1)

The central operator identity and all derived bounds are obtained by exploiting the closed-form solution of exact-fit linear regression; the manuscript must explicitly list every assumption on the feature map, task covariance, and invertibility that enables the algebraic cancellation (see the derivation of the identity). Without this, verification of the recursion is incomplete.

minor comments (2)

Abstract, lines 8-10: the phrase 'exact-fit linear regime' should appear in the opening sentence to make the scope immediately clear to readers outside the linear-regression literature.
Related-work section: the contrast with Evron et al. (2022) is stated but could be sharpened by one additional sentence on how the distributional perspective differs from their fixed-task random-order analysis.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the recommendation for minor revision. We address the major comment in detail below.

read point-by-point responses

Referee: The central operator identity and all derived bounds are obtained by exploiting the closed-form solution of exact-fit linear regression; the manuscript must explicitly list every assumption on the feature map, task covariance, and invertibility that enables the algebraic cancellation (see the derivation of the identity). Without this, verification of the recursion is incomplete.

Authors: We agree with this observation. The derivation of the central operator identity indeed relies on the closed-form expression for the exact-fit linear regressor, which in turn depends on specific assumptions regarding the feature map, the task covariances, and invertibility conditions. These assumptions are implicit in our setup but not collected in one place. To address the referee's concern, we will revise the manuscript by adding an explicit list of all such assumptions in a new paragraph at the start of the derivation section. This will facilitate verification of the recursion and the subsequent bounds. The revision will be purely expository and will not alter the technical content or results of the paper. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained algebraic identity from model assumptions

full rationale

The paper derives its central exact operator identity for the forgetting quantity directly from the closed-form solution of linear regression under the exact-fit regime with i.i.d. tasks sampled from Π, using standard linear algebra manipulations that follow from the model definition. Subsequent results (unconditional upper bound, leading asymptotic term, and convergence rate in nondegenerate cases) are obtained by further analysis of this identity without any reduction to fitted parameters, self-citations, or prior results by the same authors. The citation to evron2022catastrophic addresses a related but distinct setting (fixed tasks under random orderings) and serves only as background, not as load-bearing justification for the new identity or bounds. No steps exhibit self-definitional equivalence, fitted inputs renamed as predictions, ansatz smuggling, or uniqueness imported from self-citations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the linear exact-fit model and i.i.d. sampling from Π; no free parameters, invented entities, or non-standard axioms are mentioned in the abstract.

axioms (2)

domain assumption Tasks are generated i.i.d. from a fixed task distribution Π
Stated in abstract as the setting in which the operator identity is derived.
domain assumption Exact-fit linear regression regime
Abstract specifies this regime for the forgetting analysis.

pith-pipeline@v0.9.0 · 5484 in / 1319 out tokens · 32722 ms · 2026-05-10T13:55:37.586825+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

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doi: 10.48550/arXiv.2004.07211. URLhttps://arxiv.org/abs/2004.07211. Xufeng Cai and Jelena Diakonikolas. Last iterate convergence of incremental methods and applications in continual learning.CoRR, abs/2403.06873, 2024. doi: 10.48550/arXiv. 2403.06873. URLhttps://arxiv.org/abs/2403.06873. Ran Cheng. Context channel capacity: An information-theoretic frame...

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Daniel Goldfarb and Paul Hand

URLhttps://proceedings.mlr.press/v206/goldfarb23a.html. Daniel Goldfarb and Paul Hand. Analysis of overparameterization in continual learning under a linear model.CoRR, abs/2502.10442, 2025. doi: 10.48550/arXiv.2502.10442. URL https://arxiv.org/abs/2502.10442. Daniel Goldfarb, Itay Evron, Nir Weinberger, Daniel Soudry, and Paul Hand. The joint effect of t...

work page doi:10.48550/arxiv.2502.10442 2025

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URL https://doi.org/10.1016/j.neucom

doi: 10.1016/j.neucom.2021.10.021. URL https://doi.org/10.1016/j.neucom. 2021.10.021. Arun Mallya and Svetlana Lazebnik. PackNet: Adding multiple tasks to a single network by iterative pruning. InProceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 7765–7773, 2018. Seyed Iman Mirzadeh, Mehrdad Farajtabar, Razvan Pascanu, an...

work page doi:10.1016/j.neucom.2021.10.021 2021

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Seyed Iman Mirzadeh, Arslan Chaudhry, Huiyi Hu, Razvan Pascanu, Dilan Gorur, and Mehrdad Farajtabar

URLhttps://openreview.net/pdf?id=Fmg_fQYUejf. Seyed Iman Mirzadeh, Arslan Chaudhry, Huiyi Hu, Razvan Pascanu, Dilan Gorur, and Mehrdad Farajtabar. Wide neural networks forget less catastrophically. InProceedings of the 39th International Conference on Machine Learning. PMLR, 2022. Francesco Mori, Stefano Sarao Mannelli, and Francesca Mignacco. Optimal pro...

work page doi:10.1088/1742-5468/adf296 2022