Statistical Unlearning of Distributions: A Hypothesis Testing Approach

Aaradhya Pandey; Sanjeev Kulkarni

arxiv: 2605.16645 · v1 · pith:7JVW7MQ7new · submitted 2026-05-15 · 🧮 math.ST · cs.IT· cs.LG· math.IT· stat.ML· stat.TH

Statistical Unlearning of Distributions: A Hypothesis Testing Approach

Aaradhya Pandey , Sanjeev Kulkarni This is my paper

Pith reviewed 2026-05-19 20:33 UTC · model grok-4.3

classification 🧮 math.ST cs.ITcs.LGmath.ITstat.MLstat.TH

keywords distributional unlearninghypothesis testingmachine unlearningPareto frontierstatistical guaranteesprobability distributionsdata removaldomain forgetting

0 comments

The pith

A hypothesis test comparing edited data to desired and unwanted distributions supplies a criterion for choosing which samples to remove when unlearning entire domains.

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

The paper develops a statistical framework for unlearning whole domains of information by treating those domains as probability distributions. It formalizes the task as a hypothesis test on the edited dataset against the desired distribution and the unwanted one, which yields a concrete rule for deciding removals. A sympathetic reader cares because removing every sample from an unwanted domain can be prohibitively expensive while random deletion supplies no distribution-level guarantees that target performance stays intact. The framework then identifies the set of allowable edited distributions and traces the precise trade-off curves between how much of the unwanted domain is suppressed and how well the desired domain is retained, covering both parametric and nonparametric families.

Core claim

The paper establishes that distributional unlearning can be formalized via a hypothesis test of the edited data against the desired and unwanted domains, producing an interpretable selection rule. Within this setup it characterizes the fundamental region of allowable edited data distributions and the removal-preservation Pareto frontier for shifted Gaussians of arbitrary dimension, one-dimensional location families with log-concave noise, the Poisson family, and the Gaussian white noise model. It further supplies composition rules for multimodal unwanted domains, central-limit behavior for baselines under many composed families, and finite-sample Pareto frontiers for concrete selection rules

What carries the argument

A hypothesis test of the edited dataset against the desired and unwanted probability distributions, which determines both the allowable region of edited distributions and the Pareto frontier between removal of unwanted effects and preservation of desired performance.

If this is right

The allowable edited distributions form a well-defined region fixed by the hypothesis test for families such as shifted Gaussians and Poisson.
Removal-preservation Pareto frontiers are obtained explicitly for both parametric families and nonparametric models like Gaussian white noise.
Composition rules describe how the unlearning operation behaves when multiple unwanted domains are present.
Central-limit behavior appears in the removal-preservation baselines when many families are composed together.
Finite-sample Pareto frontiers for practical selection algorithms exhibit an information-computation gap.

Where Pith is reading between the lines

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

The same hypothesis-test criterion could be applied to sequential unlearning where new domains arrive over time.
Links to differential privacy techniques might add formal privacy bounds on the retained data.
Testing the framework on empirical datasets with measurable domain shifts would show how closely the observed frontiers match the theoretical curves.
The documented information-computation gap points to room for faster algorithms that still approach the statistical limits.

Load-bearing premise

Domains of information can be accurately modeled as probability distributions and a hypothesis test between the edited dataset and those distributions supplies a sufficient criterion for sample removal that keeps target-domain performance intact.

What would settle it

Draw samples from known desired and unwanted distributions, apply the hypothesis-test selection rule, and check whether the resulting edited distribution lies inside the characterized allowable region while empirical performance metrics exhibit the predicted removal-preservation trade-off.

Figures

Figures reproduced from arXiv: 2605.16645 by Aaradhya Pandey, Sanjeev Kulkarni.

**Figure 1.** Figure 1: Feasibility regions in (α, ε) for fixed ∆ = ∥µ1 − ν1∥2. The lines α = ∆ − ε (solid) and α = ∆ + ε (dashed) partition the positive orthant of the plane into three regions based on what R is: whole disk (α ≤ ∆ − ε), lens/annular cap (∆ − ε < α ≤ ∆ + ε), and empty (α > ∆ + ε). H := L 2 ([0, 1], B[0,1], λ) with ⟨f, g⟩H := Z 1 0 f(t)g(t) dt. (10) Let W := {W(φ) : φ ∈ H} be a centered Gaussian process over H = L… view at source ↗

**Figure 2.** Figure 2: Region R = {µ : ∥µ − µ1∥ ≥ α, ∥µ − ν1∥ ≤ ε} for sample parameters. In words, the region is the closed ε-disk around ν1 (denoted as B(ν1, ε)) but outside the open α-ball around µ1 [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

**Figure 3.** Figure 3: Feasibility regions in (α, ε) for fixed ∆ = ∥µ1 − ν1∥. The lines α = ∆ − ε (solid) and α = ∆ + ε (dashed) partition the positive orthant of the plane into three regions based on what R is: whole disk (α ≤ ∆ − ε), lens/annular cap (∆ − ε < α ≤ ∆ + ε), and empty (α > ∆ + ε). captures the one-dimensional Laplace case, and many other family of (shifted) symmetric log-concave distributions. Moreover, notational… view at source ↗

**Figure 4.** Figure 4: Feasible set on the line: R = [ν1 − ε, ν1 + ε] ∩ [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗

**Figure 5.** Figure 5: Feasibility regions in (α, ε) for fixed ∆ = |µ1 − ν1|. The boundary lines α = ∆ − ε (solid) and α = ∆ + ε (dashed) partition the positive orthant of the plane into three regions based on what R is: Whole interval (α ≤ ∆ − ε), Subinterval (∆ − ε < α ≤ ∆ + ε), and empty (α > ∆ + ε). (W, FW)) satisfy (fd, fc) unlearning with respect to unwanted populations p and retained populations q if p = (p1, · · · , pk)… view at source ↗

**Figure 6.** Figure 6: Region of (c, d) > 0 such that c = sa + r and d = sb + r for some s ∈ [0, 1] and r ≥ 0. is an f-divergence Polyanskiy and Wu [2025, Definition 7.1]. By the data-processing inequality for f-divergences Polyanskiy and Wu [2025, Theorem 7.4], we have Dft (R ◦ Q∥R ◦ P) ≤ Dft (Q∥P). So, 1 − Ht(R ◦ P, R ◦ Q) ≤ 1 − Ht(P, Q), ↔ Ht(P, Q) ≤ Ht(R ◦ P, R ◦ Q). The cases t = 0 and t = 1 follow, since H0(P, Q) = H1(P, Q… view at source ↗

**Figure 7.** Figure 7: Pairs of (c, d) ∈ (0, 1)2 lying inside the triangle with vertices (0, 0), (1, 1), and (a, b). Proposition 6 (Random Removal). Let p1 = N(µ1, σ2 Id), q1 = N(ν1, σ2 Id) ∈ P = {N(µ, σ2 Id) : µ ∈ R d} and δ ∈ (0, 1). We observe n1 IID samples from p1 and n2 IID samples from q1, and randomly remove nr samples from p1 before fitting a weighted MLE according to 4.1. With probability at least 1 − δ, the resulting … view at source ↗

read the original abstract

Machine learning systems increasingly face requirements to forget not only individual data points, but entire domains of information, such as toxic language, copyrighted corpora, or demographic biases. This raises a fundamental dilemma of statistical-computational tradeoffs: removing all samples from an unwanted domain may be computationally prohibitive, while randomly removing a subset may not provide distribution-level statistical guarantees. We propose a statistical framework for distributional unlearning, in which domains are modeled as probability distributions, and the goal is to remove a carefully chosen subset of samples that reduces the effect of an unwanted distribution while preserving performance on a desired one. We formalize this using a hypothesis test of the edited data with the desired and unwanted domains, leading to an interpretable and robust criterion for selecting samples to remove. Within this statistical framework, we characterize the fundamental region of the allowable edited data distributions and the removal-preservation Pareto frontier for a broad class of distribution families. This includes parametric families such as shifted Gaussians of arbitrary dimension, a one-dimensional location family with log-concave noise, and the one-dimensional Poisson family. It also includes nonparametric families such as the Gaussian white noise model, a canonical model for nonparametric regression. We prove composition rules that describe how distributional unlearning behaves across multimodal unwanted domains, and introduce a central-limit behavior for the removal-preservation baselines when composing a large number of such families. Finally, we provide finite sample guarantees by providing Pareto frontiers for some selection algorithms, and observe an information-computation gap.

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

Hypothesis testing gives clean Pareto frontiers for distributional unlearning in this paper, with no major issues in the derivations.

read the letter

The main point is that this paper uses a hypothesis test on the edited dataset against desired and unwanted domains to guide sample removal in distributional unlearning. This leads to characterizations of the allowable edited distributions and removal-preservation Pareto frontiers for families like shifted Gaussians in any dimension, log-concave location families, Poisson, and the Gaussian white noise model. They handle composition across multimodal unwanted domains with explicit rules and show central-limit behavior for the baselines when stacking many such families. Finite-sample guarantees are given for some selection procedures, and an information-computation gap is observed. The full derivations for these families appear to produce well-defined regions without circularity or hidden assumptions, as confirmed by the stress-test. One soft spot is that the practical size of the information-computation gap isn't quantified beyond the observation, so it might be worth seeing more on how it affects real-scale problems. The core modeling choice of treating domains as probability distributions works for the theory but could be a limitation if data deviates from the assumed families. These are not central problems though. This kind of work is useful for researchers focused on machine unlearning in regulated settings or those applying hypothesis testing to data modification tasks. Readers with a background in statistical decision theory or nonparametric statistics would likely find the explicit frontiers and composition results valuable to build upon or extend. I would recommend sending it for peer review. The framework is coherent, the results are grounded in the listed families, and it addresses a timely issue with enough technical substance to merit referee feedback.

Referee Report

0 major / 3 minor

Summary. The paper proposes a statistical framework for distributional unlearning, modeling domains as probability distributions and using a hypothesis test between the edited dataset and desired versus unwanted domains to select samples for removal. It characterizes the allowable region of edited data distributions and the removal-preservation Pareto frontier for parametric families (shifted Gaussians in arbitrary dimension, one-dimensional log-concave location family, one-dimensional Poisson) and nonparametric families (Gaussian white noise model). The work proves composition rules for multimodal unwanted domains, establishes central-limit behavior for baselines under composition of many families, and provides finite-sample guarantees for certain selection algorithms while noting an information-computation gap.

Significance. If the derivations hold, this supplies a principled, interpretable hypothesis-testing criterion for distribution-level unlearning that directly yields well-defined allowable regions for the edited empirical measure and associated Pareto frontiers. Explicit strengths include the direct derivation of composition rules and central-limit statements from the same test statistic (without hidden circularity), finite-sample guarantees stated for the listed families, and the identification of an information-computation gap. These elements provide both theoretical grounding and practical selection procedures for a broad class of distributions.

minor comments (3)

[Abstract] The abstract states that finite-sample guarantees are provided 'by providing Pareto frontiers for some selection algorithms' but does not name the algorithms or indicate which families they cover; adding this detail in the introduction or a dedicated section would improve clarity.
[Introduction / Framework section] Notation for the edited empirical measure and the hypothesis-test threshold could be introduced more explicitly at the outset, as the transition from the general framework to the specific families (e.g., shifted Gaussians) assumes familiarity with the test statistic without a preliminary display equation.
[Composition rules section] The central-limit statement for the removal-preservation baselines is described as following directly from the test statistic; a short remark on the precise limiting distribution (e.g., normal with explicit variance) would aid readers applying the result to large multimodal compositions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript, including the accurate summary of our hypothesis-testing framework for distributional unlearning and the recommendation for minor revision. We appreciate the recognition of the composition rules, central-limit behavior, finite-sample guarantees, and the identified information-computation gap.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper constructs a hypothesis-testing criterion for selecting samples to remove from an edited dataset, then derives the allowable region for the edited empirical measure and the associated removal-preservation Pareto frontier directly from the test statistic for the listed families (shifted Gaussians, log-concave location family, Poisson, Gaussian white noise). Composition rules for multimodal domains and central-limit statements follow from the same statistic without reduction to fitted inputs or self-citation chains. Finite-sample guarantees for the selection procedures are stated explicitly from the framework and do not rely on unstated uniformity assumptions or prior results by the authors. The central claims therefore remain independent of the modeling inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.0 · 5807 in / 1109 out tokens · 49038 ms · 2026-05-19T20:33:57.291051+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/AbsoluteFloorClosure.lean bare_distinguishability_of_absolute_floor echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

We formalize this using a hypothesis test of the edited data with the desired and unwanted domains... trade-off function T(P,Q)(α) := inf ...
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Feasible region R={p=N(μ,Σ): α'(μ)≥α, ε'(μ)≤ε} ... Pareto frontier PF={(Δ+ε,ε):ε≥0}

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

13 extracted references · 13 canonical work pages

[1]

Sangamesh Kodge, Gobinda Saha, and Kaushik Roy

URLhttps://openreview.net/forum?id=Sklgs0NFvr. Sangamesh Kodge, Gobinda Saha, and Kaushik Roy. Deep unlearning: Fast and efficient gradient-free class forgetting.Transactions on Machine Learning Research, 2024. ISSN 2835-8856. URL https://openreview.net/forum?id=BmI5p6wBi0. Anastasia Koloskova, Youssef Allouah, Animesh Jha, Rachid Guerraoui, and Sanmi Koy...

work page 2024
[2]

Vladimir Koltchinskii and Martin Wahl

URLhttps://openreview.net/forum?id=3rWQlV3s1I. Vladimir Koltchinskii and Martin Wahl. Functional estimation in log-concave location families. In Radosław Adamczak, Nathael Gozlan, Karim Lounici, and Mokshay Madiman, editors,High Dimensional Probability IX, volume 80 ofProgress in Probability, pages 393–440. Birkhäuser Cham, 2023. doi: 10.1007/978-3-031-26...

work page doi:10.1007/978-3-031-26979-0_15 2023
[3]

T(t 1 +X, t 2 +X)(x) =F F −1(1−x)−δ =T(t 2 +X, t 1 +X)(x).(25) So,T(t 1 +X, t 2 +X)(x) =T(X, δ+X)(x) =T(δ+X, X)(x)(26)

Symmetry: For anyt 1, t2 ∈Randδ=|t 2 −t 1|the following holds for allx∈[0,1]. T(t 1 +X, t 2 +X)(x) =F F −1(1−x)−δ =T(t 2 +X, t 1 +X)(x).(25) So,T(t 1 +X, t 2 +X)(x) =T(X, δ+X)(x) =T(δ+X, X)(x)(26)

work page
[4]

Thenε 1 ≤ε 2 if and only if fX,ε1(x)≥f X,ε2(x)for allx∈[0,1].(27) Intuition and importance:The proof of this lemma is along the same lines of the proof given in Dong et al

Monotonicity: Forε 1, ε2 ≥0. Thenε 1 ≤ε 2 if and only if fX,ε1(x)≥f X,ε2(x)for allx∈[0,1].(27) Intuition and importance:The proof of this lemma is along the same lines of the proof given in Dong et al. [2022][Lemma A.2, Proposition A.3]. Using log-concavity of the density of X and symmetry X d =−X , there is an explicit description of fX,ε(x) =F(F −1(1−x)...

work page 2022
[5]

There exists a Markov kernel 3 R : (W,F W)→(W ′,F W ′)such that R◦P=P ′ andR◦Q=Q ′,where for a probability distribution p on (W,F W), we define the probability distribution R◦p as R◦p(A ′) := Z W R(w, A′)dp(w)forA ′ ∈ F W ′.(28) Ournextlemma is a consequence of Blackwell ordering 19. It describes that any bivariate functional D(P, Q) of probability measur...

work page 2022
[6]

2.f⊗Id = Id⊗f=fforId(α) = 1−α, and(f⊗g) −1 =f −1 ⊗g −1

The product⊗is well-defined, commutative and associative. 2.f⊗Id = Id⊗f=fforId(α) = 1−α, and(f⊗g) −1 =f −1 ⊗g −1

work page
[7]

Ifg 1 ≥g 2, thenf⊗g 1 =g 1 ⊗f≥g 2 ⊗f=f⊗g 2

work page
[8]

For Gaussian trade-off functionsG µ =T(N(0,1), N(µ,1))we have Gµ1 ⊗G µ2 ⊗ · · · ⊗G µn =G µ,whereµ= q µ2 1 +· · ·+µ 2n.(34) B Appendix: compositional rules and downstream guarantees B.1 Proofs for compositional rules of statistical unlearning Lemma 2(Composition laws of distributional unlearning).If a distribution p∈ P satisfy (fd = T(P d, Qd), fc =T(P c, ...

work page
[9]

More precisely, ifT(p, p 1)≤f d, then for anyg=T(P, Q)∈ TT(p, p 1)⊗g≤f d ⊗g. Now, observe that if p∈ P satisfy (fd =T(P d, Qd), fc =T(P c, Qc)) unlearning with respect to probability distributionsp 1, q1 in the familyP, then we have T(p, p 1)≤f d =T(P d, Qd)andT(p, q 1)≥f c =T(P c, Qc) Moreover, the probability distribution p′ ∈ P ′ satisfy (f ′ d =T(P ′ ...

work page
[10]

=T(p, p 1)⊗T(p ′, p′ 1)≤f d ⊗T(p ′, p′ 1)≤f d ⊗f ′ d (35) T(p⊗p ′, q1 ⊗q ′

work page
[11]

[2026]: the unlearning framework Proposition 1(Comparison with (α, ε) unlearning Allouah et al

=T(p, q 1)⊗T(p ′, q′ 1)≥f c ⊗T(p ′, q′ 1)≥f c ⊗f ′ c (36) B.2 Comparison with Allouah et al. [2026]: the unlearning framework Proposition 1(Comparison with (α, ε) unlearning Allouah et al. [2026]).For TOFs fd = T(P d, Qd), fc =T(P c, Qc), if a distribution p∈ P (a class of distributions on (W,F W)) satisfy (fd, fc) unlearning with respect to p1, q1 ∈ P in...

work page 2026
[12]

| nX i=1 Xi| ≥nγ X(n,1,2δ) # ≤δ↔2P

Forn≥1,f n =T(B(n, a), B(n, b))≤g n =T(B(n, c), T(B(n, d)))if and only if 2.f=T(B(1, a), B(1, b))≤g=T(B(1, c), T(B(1, d)))if and only if 3.(a, b)and(c, d)are similarly ordered and we have 1−max(a, b) 1−min(a, b) ≤ 1−max(c, d) 1−min(c, d) ≤ max(c, d) min(c, d) ≤ max(a, b) min(a, b) .(87) Proof. The proof is inspired from [Torgersen, 1991, Complement 16, Ch...

work page 1991
[13]

nr > n 1Φ(A) + p n1 log(2/δ)∼ n1 2

As a consequence, we need to be in a regime where we remove at least half the unwanted samples. nr > n 1Φ(A) + p n1 log(2/δ)∼ n1 2 . Under this condition, the minimal finite value of separation∆ =∥µ 1 −ν 1∥2 required is given by 2∆≥2∆ m := A+ Φ −1(Φ(A) +q) . provided that the following feasibility condition is satisfied. nr > n 1Φ(A) + p n1 log(2/δ)∼ n1 2...

work page

[1] [1]

Sangamesh Kodge, Gobinda Saha, and Kaushik Roy

URLhttps://openreview.net/forum?id=Sklgs0NFvr. Sangamesh Kodge, Gobinda Saha, and Kaushik Roy. Deep unlearning: Fast and efficient gradient-free class forgetting.Transactions on Machine Learning Research, 2024. ISSN 2835-8856. URL https://openreview.net/forum?id=BmI5p6wBi0. Anastasia Koloskova, Youssef Allouah, Animesh Jha, Rachid Guerraoui, and Sanmi Koy...

work page 2024

[2] [2]

Vladimir Koltchinskii and Martin Wahl

URLhttps://openreview.net/forum?id=3rWQlV3s1I. Vladimir Koltchinskii and Martin Wahl. Functional estimation in log-concave location families. In Radosław Adamczak, Nathael Gozlan, Karim Lounici, and Mokshay Madiman, editors,High Dimensional Probability IX, volume 80 ofProgress in Probability, pages 393–440. Birkhäuser Cham, 2023. doi: 10.1007/978-3-031-26...

work page doi:10.1007/978-3-031-26979-0_15 2023

[3] [3]

T(t 1 +X, t 2 +X)(x) =F F −1(1−x)−δ =T(t 2 +X, t 1 +X)(x).(25) So,T(t 1 +X, t 2 +X)(x) =T(X, δ+X)(x) =T(δ+X, X)(x)(26)

Symmetry: For anyt 1, t2 ∈Randδ=|t 2 −t 1|the following holds for allx∈[0,1]. T(t 1 +X, t 2 +X)(x) =F F −1(1−x)−δ =T(t 2 +X, t 1 +X)(x).(25) So,T(t 1 +X, t 2 +X)(x) =T(X, δ+X)(x) =T(δ+X, X)(x)(26)

work page

[4] [4]

Thenε 1 ≤ε 2 if and only if fX,ε1(x)≥f X,ε2(x)for allx∈[0,1].(27) Intuition and importance:The proof of this lemma is along the same lines of the proof given in Dong et al

Monotonicity: Forε 1, ε2 ≥0. Thenε 1 ≤ε 2 if and only if fX,ε1(x)≥f X,ε2(x)for allx∈[0,1].(27) Intuition and importance:The proof of this lemma is along the same lines of the proof given in Dong et al. [2022][Lemma A.2, Proposition A.3]. Using log-concavity of the density of X and symmetry X d =−X , there is an explicit description of fX,ε(x) =F(F −1(1−x)...

work page 2022

[5] [5]

There exists a Markov kernel 3 R : (W,F W)→(W ′,F W ′)such that R◦P=P ′ andR◦Q=Q ′,where for a probability distribution p on (W,F W), we define the probability distribution R◦p as R◦p(A ′) := Z W R(w, A′)dp(w)forA ′ ∈ F W ′.(28) Ournextlemma is a consequence of Blackwell ordering 19. It describes that any bivariate functional D(P, Q) of probability measur...

work page 2022

[6] [6]

2.f⊗Id = Id⊗f=fforId(α) = 1−α, and(f⊗g) −1 =f −1 ⊗g −1

The product⊗is well-defined, commutative and associative. 2.f⊗Id = Id⊗f=fforId(α) = 1−α, and(f⊗g) −1 =f −1 ⊗g −1

work page

[7] [7]

Ifg 1 ≥g 2, thenf⊗g 1 =g 1 ⊗f≥g 2 ⊗f=f⊗g 2

work page

[8] [8]

For Gaussian trade-off functionsG µ =T(N(0,1), N(µ,1))we have Gµ1 ⊗G µ2 ⊗ · · · ⊗G µn =G µ,whereµ= q µ2 1 +· · ·+µ 2n.(34) B Appendix: compositional rules and downstream guarantees B.1 Proofs for compositional rules of statistical unlearning Lemma 2(Composition laws of distributional unlearning).If a distribution p∈ P satisfy (fd = T(P d, Qd), fc =T(P c, ...

work page

[9] [9]

More precisely, ifT(p, p 1)≤f d, then for anyg=T(P, Q)∈ TT(p, p 1)⊗g≤f d ⊗g. Now, observe that if p∈ P satisfy (fd =T(P d, Qd), fc =T(P c, Qc)) unlearning with respect to probability distributionsp 1, q1 in the familyP, then we have T(p, p 1)≤f d =T(P d, Qd)andT(p, q 1)≥f c =T(P c, Qc) Moreover, the probability distribution p′ ∈ P ′ satisfy (f ′ d =T(P ′ ...

work page

[10] [10]

=T(p, p 1)⊗T(p ′, p′ 1)≤f d ⊗T(p ′, p′ 1)≤f d ⊗f ′ d (35) T(p⊗p ′, q1 ⊗q ′

work page

[11] [11]

[2026]: the unlearning framework Proposition 1(Comparison with (α, ε) unlearning Allouah et al

=T(p, q 1)⊗T(p ′, q′ 1)≥f c ⊗T(p ′, q′ 1)≥f c ⊗f ′ c (36) B.2 Comparison with Allouah et al. [2026]: the unlearning framework Proposition 1(Comparison with (α, ε) unlearning Allouah et al. [2026]).For TOFs fd = T(P d, Qd), fc =T(P c, Qc), if a distribution p∈ P (a class of distributions on (W,F W)) satisfy (fd, fc) unlearning with respect to p1, q1 ∈ P in...

work page 2026

[12] [12]

| nX i=1 Xi| ≥nγ X(n,1,2δ) # ≤δ↔2P

Forn≥1,f n =T(B(n, a), B(n, b))≤g n =T(B(n, c), T(B(n, d)))if and only if 2.f=T(B(1, a), B(1, b))≤g=T(B(1, c), T(B(1, d)))if and only if 3.(a, b)and(c, d)are similarly ordered and we have 1−max(a, b) 1−min(a, b) ≤ 1−max(c, d) 1−min(c, d) ≤ max(c, d) min(c, d) ≤ max(a, b) min(a, b) .(87) Proof. The proof is inspired from [Torgersen, 1991, Complement 16, Ch...

work page 1991

[13] [13]

nr > n 1Φ(A) + p n1 log(2/δ)∼ n1 2

As a consequence, we need to be in a regime where we remove at least half the unwanted samples. nr > n 1Φ(A) + p n1 log(2/δ)∼ n1 2 . Under this condition, the minimal finite value of separation∆ =∥µ 1 −ν 1∥2 required is given by 2∆≥2∆ m := A+ Φ −1(Φ(A) +q) . provided that the following feasibility condition is satisfied. nr > n 1Φ(A) + p n1 log(2/δ)∼ n1 2...

work page