Pith. sign in

REVIEW 4 minor 54 references

Under central differential privacy, the minimax L1 separation radius for two-sample testing of Hölder-smooth densities is exactly the maximum of the classical nonprivate rate and three distinct privacy barriers.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Under central differential privacy, the sharp L1 separation radius for two-sample testing of Hölder-smooth densities is the maximum of the classical rate and three privacy barriers, and adapting to unknown smoothness costs only a log-log factor.

T0 review reviewed 2026-07-31 challenge →

load-bearing objection Real deal: if the long appendix holds up, this settles the central-DP version of the smooth two-sample testing boundary with a four-term rate and a clean phase split at s = d/4.

arxiv 2607.23974 v1 pith:ZI4EHACB submitted 2026-07-27 math.ST stat.TH

Sharp Minimax Rates for Smooth Two-Sample Testing under Central Differential Privacy

Ilmun Kim This is my paper
classification math.ST stat.TH MSC 62G1062G2062C20
keywords central differential privacytwo-sample testingminimax separation radiusHölder smooth densitiesL1 distancebounded histogramsadaptive hypothesis testingphase transition
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

This paper works out the exact separation boundary for the most basic privacy-aware comparison of two samples: given independent samples from two smooth densities on the unit cube, decide whether the densities are equal or differ by at least r in total variation, while releasing only an ε-differentially private decision. The answer, for Nε≥1, is that the smallest detectable L1 distance is the maximum of four terms, not a single 'privacy makes N smaller' correction: the classical nonprivate smooth-testing rate plus three separate privacy barriers that arise from within-cell information, private transport on the hypercube, and a linear coupling floor. Which barrier dominates depends on the smoothness-to-dimension ratio s/d and on how the privacy budget scales with N, yielding a phase diagram with three or four regimes. The paper constructs tests that attain this boundary with finite-sample type-I control, including a fully averaged split-averaged permutation-calibrated statistic, and proves each of the four terms unavoidable. It also shows that when smoothness is unknown, a multiscale private test pays exactly an iterated-logarithmic penalty in the classical term and nothing more.

Core claim

The central claim is that the minimax L1 separation radius for Hölder-smooth two-sample testing under central differential privacy is exactly r*(N,ε) ≍ N^{-2s/(4s+d)} ∨ (N√ε)^{-2s/(2s+d)} ∨ (N^{3/2}ε)^{-2s/(4s+d)} ∨ (Nε)^{-1}, with all four terms necessary. The first term is the rate achievable without privacy; the other three are genuine privacy-induced barriers. The construction that attains the rate bins the data into histograms, applies a private discrete two-sample test to the binned counts, and chooses the resolution to balance bias, sampling noise, and Laplace noise. The lower-bound proof isolates four mechanisms—the classical smooth hypercube, within-cell sign perturbations, a privac

What carries the argument

The central object is the bounded-histogram subclass P_{k,M} of discrete distributions with all cell probabilities at most M/k, which smooth bounded densities induce after binning. On this subclass, the unrestricted discrete closeness-testing bound loses its heavy-element term k^{2/3}/τ^{4/3}; the split statistic—comparing two cross-sample absolute differences against two within-sample ones—has global sensitivity 4 and is calibrated by a Monte Carlo permutation p-value with Laplace noise, giving finite-sample type-I control. A new transport inequality for ε-DP tests, relating the increase in rejection probability under an absolutely continuous mixture to ε, the sample size, and the KL diverg

Load-bearing premise

The load-bearing premise is the bounded-histogram closeness lemma: uniformly over all pairs of binned distributions with cell masses at most M/k, the split statistic's expectation is at least c·min{nτ, n²τ²/k, n^{3/2}τ²/√k} and its variance is at most 64Mn²/k; if that bound fails in any regime, the second and third privacy terms of the rate change.

What would settle it

Take a small bounded-histogram example, e.g. k=4, M=2, p=(1/2,1/4,1/4,0), q=(1/2-τ/4,1/4+τ/4,1/4,0), and compute the exact expected split statistic D_n(p,q) from its integral representation D_n=(2/π)∫|u_p(t)-u_q(t)|²/t² dt for n=10 and τ=0.1. If D_n is smaller than (1/16π)min{n²τ²/k, nτ, n^{3/2}τ²/√k} by a constant factor, the binomial comparison lemma fails and the upper bound collapses. Similarly, simulation of the private permutation test at the phase boundary could check empirically whether the (N^{3/2}ε)^{-2s/(4s+d)} term is in fact necessary.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • At separation r, the optimal sample size under central DP is the maximum of the four inverse rates: r^{-(4s+d)/(2s)}, ε^{-1/2}r^{-(2s+d)/(2s)}, ε^{-2/3}r^{-(4s+d)/(3s)}, and (εr)^{-1}.
  • Privacy-budget scaling ε=N^{-α} gives a four-regime phase diagram when s<d/4 and a three-regime diagram when s≥d/4; the (N√ε) barrier is active only in the rough or high-dimensional case.
  • Binning smooth densities removes the heavy-element nonprivate cost of unrestricted private closeness testing, so the binned problem's nonprivate cost is just √k/τ².
  • The adaptive multiscale private test achieves the fixed-smoothness privacy terms and pays exactly a (log log N)^{s/(4s+d)} penalty in the classical term, with a matching lower bound showing this cost is unavoidable.
  • A split-averaged implementation evaluates the statistic in O(NX+NY) time from the contingency table and preserves the same four-term rate with finite-sample validity.

Where Pith is reading between the lines

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

  • Editorial extension: because the adaptation penalty lands only on the classical term, private two-sample testing may be more robust to unknown smoothness than its nonprivate counterpart; a practitioner who already pays the privacy cost could get adaptation nearly for free in privacy-dominated regimes.
  • Editorial extension: the same bounded-histogram split-statistic mechanism should transfer to private goodness-of-fit testing against a known density, yielding an analogous four-term boundary; this is the one-sample version of the reduction the paper already uses for its lower bounds.
  • Editorial extension: the transport inequality could be reused as a general lower-bound tool for other private nonparametric testing problems whose alternatives are Hamming-couplable mixtures, such as private signal detection in Gaussian white noise; a testable check is whether it recovers the known phase structure in that model.
  • Editorial extension: the sensitivity invariance under taking maxima suggests that multiscale private tests can search over many resolutions without splitting the privacy budget; this observation may extend beyond density testing to any bounded-sensitivity family of statistics.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper studies minimax L1 two-sample testing of Hölder-smooth densities under central differential privacy. It claims that for Nε≥1 the minimax separation radius is the maximum of four terms: the classical nonprivate rate N^{-2s/(4s+d)}, two intermediate privacy rates (N√ε)^{-2s/(2s+d)} and (N^{3/2}ε)^{-2s/(4s+d)}, and the linear privacy barrier (Nε)^{-1}. The upper bound bins the data, applies a private permutation-calibrated split statistic on the resulting bounded histograms, and optimizes the bin width; a Rao–Blackwellized variant and an adaptive multiscale version are also developed. The lower bounds reduce the problem to one-sample goodness-of-fit against uniform and derive the four terms via smooth hypercube perturbations, within-cell DP coupling, a new transport inequality, and a fixed-perturbation coupling. The paper also proves that adaptation to unknown smoothness costs exactly an iterated-logarithmic factor, even in the nonprivate case.

Significance. If correct, this settles the exact minimax boundary for smooth two-sample testing under central DP, giving a sharp four-term phase diagram. The paper is unusually complete: the four lower-bound mechanisms, the bandwidth optimizations, and the finite-sample permutation calibration are all worked out in detail. The bounded-histogram refinement of private closeness testing and the new transport inequality are substantive technical contributions. The adaptive lower bound is also a genuine contribution that does not depend on privacy. I found no concrete mathematical failure in the load-bearing arguments: the flat-histogram closeness engine (Lemmas S.10–S.13, Propositions 5 and 9) is internally consistent, and the phase boundaries and bandwidth algebra check out. The principal residual risk is verification burden rather than an identified error: there are no machine-checked proofs and the upper bound leans on the author's own Kim–Schrab (2026) private permutation theorem, stated but not proved in this manuscript.

minor comments (4)
  1. [Appendix A.4, Lemma S.10] The displayed characteristic-function identity writes Re(u_p(t)u_q(t)) in the integrand; it should be Re(u_p(t)\overline{u_q(t)}). The proof immediately uses |u_p-u_q|^2, i.e. the correct expression, so this is a notational slip rather than a mathematical one, but it should be corrected to avoid misleading readers.
  2. [Lemma S.7 and Propositions 5/9] The finite-sample level and power guarantees rely on the private permutation theorem of Kim and Schrab (2026), and in places on 'Theorem 4' of that paper, without stating the exact theorem or its conditions. Since this is a load-bearing external result and also a self-citation, the authors should either include a precise statement of the theorem used or give a self-contained proof in the appendix.
  3. [Section 6, Algorithm 3] The adaptive test is stated only for even N=2n, and Theorem 12 likewise restricts to even N. This is a presentation limitation, but the paper should either state the extension to odd N or explicitly note that N can be replaced by 2⌊N/2⌋ at the cost of constants.
  4. [Section 7, Figure 2] The numerical experiments are helpful illustrations, but the caption and text should more explicitly warn that the simulation settings are not designed to verify the minimax exponents; the current wording already says this, but it is worth keeping prominent given the temptation to read the plots as rate evidence.

Circularity Check

0 steps flagged

No significant circularity: the four-term rate and its upper/lower bounds are independently derived; the main self-citations are external black-box theorems with assumptions verified in this paper.

full rationale

The central derivation is not circular. The upper bound is built from explicit moment bounds (Lemmas S.10, S.12, S.13, S.19) for the split and Rao–Blackwellized statistics, and the binning bias is controlled by Lemma S.6; the four privacy terms come from balancing those proved discrete costs, not from assuming the target rate. The lower bounds are proved separately for each term: the classical hypercube construction (Prop. S.20), the within-cell privacy coupling (Prop. S.23), the transport inequality (Lemma S.9) applied to a hypercube mixture (Prop. S.27), and the fixed-perturbation linear barrier (Prop. S.28). None of these reduces to the upper bound by construction. The adaptive result (Theorem 12) and its matching lower bound (Theorem 13) are also substantive: the lower bound is a nonprivate multiscale Ingster mixture with cross-scale Gram estimates, independent of privacy machinery. The paper does cite the author's own work: Lemma S.7 restates and uses the Kim–Schrab private permutation theorem, and it is load-bearing for the finite-sample level guarantee. However, this is a general external theorem whose stated assumptions (bounded sensitivity and exchangeability under the null) are checked here (Lemma S.11 and the proof of Prop. 4), so the citation is independent support rather than a circular reduction. The mention of Schrab et al. (2026) is only an analogy for the two-batch aggregation idea and is not load-bearing. I found no fitted parameter renamed as a prediction, no self-definitional equivalence, and no imported uniqueness claim. The paper is self-contained against the standard nonprivate smooth-testing benchmark and against the public discrete closeness-testing results it refines.

Axiom & Free-Parameter Ledger

0 free parameters · 9 axioms · 0 invented entities

The central claim rests on standard minimax-testing machinery (bumps, hypercubes, Fano/Le Cam-type reductions), a handful of imported cited theorems (Kim–Schrab permutation calibration; Acharya et al.'s DP-coupling and Bernoulli-mixture lemmas), one new lemma proved in the paper (the transport inequality S.9), and domain assumptions (Nε ≥ 1; D_d^s(L) ⊂ F_d^s(L,L)) that are stated. There are no fitted free parameters and no invented entities. The heaviest upstream imports are the self-cited Kim–Schrab theorem powering all upper bounds and the Acharya et al. coupling lemmas powering two of four lower bounds.

axioms (9)
  • standard math Hölder density class D_d^s(L) as defined in Section 2.1 (sup-norm and Hölder-seminorm bounds on derivatives up to order q = ⌈s⌉−1)
    The model class of the problem; standard function-space definition. Used in Section 2.1.
  • domain assumption The Nε ≥ 1 regime restriction in Theorem 1; for Nε < 1 the radius is constant order since (Nε)^{-1} ≥ 1 and L1 distances are at most 2
    Scope of the main theorem; stated in Section 3.1 and in the proof of Corollary 2.
  • domain assumption D_d^s(L) ⊂ F_d^s(L, L) with envelope M = L, because the Hölder norm includes the sup-norm bound
    Lets the bounded-class upper bound (Theorem 6/Corollary 7) transfer to D_d^s(L); used in Corollary 7.
  • domain assumption Private Monte Carlo permutation theorem of Kim and Schrab (2026, Theorems 1–2 and 4), including finite-sample level control and the moment-condition power guarantee
    Lemma S.7 is the calibration engine of every upper bound (Algorithms 1–3); imported from the author's own published JASA paper. The application-specific sensitivity bound (Lemma S.11) is proved in this paper.
  • standard math DP-coupling lemma of Acharya et al. (2018): any ε-DP test with type-I error γ and type-II error β against Q requires E[d_H] ≥ ε^{-1} log(1/(γ+β)) under any (P,Q)-coupling
    Lemma S.8; powers the Regime II (within-cell) and Regime IV (linear floor) lower bounds; proof included.
  • standard math Bernoulli-mixture coupling lemma of Acharya et al. (2018, Lemma 12): Rademacher vs bias-δ mixture on t coordinates couples with E[d_H] ≤ 4(t²−t)δ²
    Lemma S.26; converts within-cell sign mixtures into expected-Hamming cost in Regime II.
  • standard math Existence of C∞ bumps ψ with zero integral, unit L2 norm, and finite Hölder norm (Lemma S.21)
    Base ingredient of the lower-bound hypercube constructions; proved in the paper.
  • standard math Standard concentration: McDiarmid, Efron–Stein (Lemma S.1), sampling-without-replacement and multislice bounded differences (Lemmas S.2–S.3), and the entropy variational formula EQ[F] ≤ KL(Q∥P) + log EP e^F
    Used in the upper-bound moment arguments and in the transport inequality Lemma S.9; cited to Boucheron, Lugosi & Massart (2013).
  • domain assumption In the GOF-to-two-sample reduction (Lemma 10), the auxiliary uniform sample is internal randomness of the mechanism, not private data, so DP of the two-sample test transfers
    Basis of the lower-bound reduction; stated and proved in Appendix B.12.

reviewed 2026-07-31 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Sharp Minimax Rates for Smooth Two-Sample Testing under Central Differential Privacy." pith.science (2026). https://pith.science/paper/ZI4EHACB

@misc{pith2026260723974,
  author       = {Pith},
  title        = {Pith review of: Sharp Minimax Rates for Smooth Two-Sample Testing under Central Differential Privacy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZI4EHACB}},
  note         = {Machine review of arXiv:2607.23974}
}
Share X Bluesky LinkedIn Reddit HN
abstract

We establish sharp minimax limits for two-sample testing of H\"older-smooth densities under central differential privacy. Given two independent samples, the goal is to decide whether the underlying distributions are identical or separated in $L_1$ distance, while releasing only an $\varepsilon$-differentially private decision. We show that privacy changes the classical smooth-testing boundary through multiple regimes: the optimal separation radius is the maximum of four terms, consisting of the classical nonprivate rate and three distinct privacy-induced barriers. Which barrier is active depends on the privacy budget and the smoothness-to-dimension ratio, yielding a sharp phase diagram. Our upper bound discretizes the samples, applies a private discrete two-sample test to the resulting histograms, and chooses the bin resolution to balance approximation bias, sampling fluctuations, and privacy noise. The procedure also admits a permutation-calibrated implementation with finite-sample type~I error control. For the lower bounds, we combine smooth perturbation constructions with privacy-specific coupling and transport inequalities, showing that all four terms are unavoidable. Finally, when the smoothness is unknown, we develop a multiscale private test that attains the optimal adaptive rate and prove a matching lower bound. Adaptation costs exactly an iterated-logarithmic factor, and this cost appears only in the classical nonprivate term.

Figures

Figures reproduced from arXiv: 2607.23974 by Ilmun Kim.

Figure 1
Figure 1. Figure 1: Phase structure of the minimax rate r ∗ 2samp(N, ε) ≍ N −ρ(α) under ε = N −α. The bold curve is the lower envelope ρ(α) = mini ρi(α). Thin dashed, dash-dotted, or dotted curves show the individual exponent functions, and shaded patterned regions indicate the active regimes. Left (s = 1, d = 8, s < d/4): all four exponents attain the envelope, producing four phases with transitions at α = 1 3 , 3 4 , 9 10 .… view at source ↗
Figure 2
Figure 2. Figure 2: Finite-sample behavior of the private permutation tests. Panel A reports empirical [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Finite-sample behavior of the adaptive multiscale test. Panel A reports empirical type I [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

54 extracted references · 15 canonical work pages

  1. [1]

    Differentially private testing of identity and closeness of discrete distributions

    Jayadev Acharya, Ziteng Sun, and Huanyu Zhang. Differentially private testing of identity and closeness of discrete distributions. In Advances in Neural Information Processing Systems 31 ( NeurIPS 2018) , pages 6878--6891, 2018

    2018

  2. [2]

    Canonne, Cody Freitag, and Himanshu Tyagi

    Jayadev Acharya, Cl\' e ment L. Canonne, Cody Freitag, and Himanshu Tyagi. Test without trust: Optimal locally private distribution testing. In Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics ( AISTATS 2019) , volume 89 of Proceedings of Machine Learning Research, pages 2067--2076. PMLR, 2019

    2019

  3. [3]

    Canonne, Cody Freitag, Ziteng Sun, and Himanshu Tyagi

    Jayadev Acharya, Cl\' e ment L. Canonne, Cody Freitag, Ziteng Sun, and Himanshu Tyagi. Inference under information constraints III : Local privacy constraints. IEEE Journal on Selected Areas in Information Theory , 2 0 (1): 0 253--267, 2021 a . doi:10.1109/JSAIT.2021.3053569

    arXiv 2021

  4. [4]

    Differentially private A ssouad, F ano, and L e C am

    Jayadev Acharya, Ziteng Sun, and Huanyu Zhang. Differentially private A ssouad, F ano, and L e C am. In Proceedings of the 32nd International Conference on Algorithmic Learning Theory ( ALT ) , volume 132 of Proceedings of Machine Learning Research, pages 48--78. PMLR, 2021 b

    2021

  5. [5]

    Differentially private identity and equivalence testing of discrete distributions

    Maryam Aliakbarpour, Ilias Diakonikolas, and Ronitt Rubinfeld. Differentially private identity and equivalence testing of discrete distributions. In Proceedings of the 35th International Conference on Machine Learning ( ICML ) , volume 80 of Proceedings of Machine Learning Research, pages 169--178. PMLR, 2018

    2018

  6. [6]

    Kane, and Ronitt Rubinfeld

    Maryam Aliakbarpour, Ilias Diakonikolas, Daniel M. Kane, and Ronitt Rubinfeld. Private testing of distributions via sample permutations. In Advances in Neural Information Processing Systems 32 ( NeurIPS 2019) , pages 10877--10888, 2019

    2019

  7. [7]

    Remember the curse of dimensionality: The case of goodness-of-fit testing in arbitrary dimension

    Ery Arias-Castro, Bruno Pelletier, and Venkatesh Saligrama. Remember the curse of dimensionality: The case of goodness-of-fit testing in arbitrary dimension. Journal of Nonparametric Statistics, 30 0 (2): 0 448--471, 2018. doi:10.1080/10485252.2018.1435875

    arXiv 2018

  8. [8]

    Differentially private Kolmogorov--Smirnov -type tests

    Jordan Awan and Yue Wang. Differentially private Kolmogorov--Smirnov -type tests. Electronic Journal of Statistics, 19 0 (1): 0 718--744, 2025. doi:10.1214/25-EJS2349

    work page doi:10.1214/25-ejs2349 2025

  9. [9]

    Hypothesis testing for densities and high-dimensional multinomials: Sharp local minimax rates

    Sivaraman Balakrishnan and Larry Wasserman. Hypothesis testing for densities and high-dimensional multinomials: Sharp local minimax rates. The Annals of Statistics, 47 0 (4): 0 1893--1927, 2019. doi:10.1214/18-AOS1729

    work page doi:10.1214/18-aos1729 1927

  10. [10]

    Rina Foygel Barber and John C. Duchi. Privacy and statistical risk: Formalisms and minimax bounds. arXiv preprint arXiv:1412.4451, 2014

    Pith/arXiv arXiv 2014

  11. [11]

    Smith, and Patrick White

    Tu g kan Batu, Lance Fortnow, Ronitt Rubinfeld, Warren D. Smith, and Patrick White. Testing closeness of discrete distributions. Journal of the ACM, 60 0 (1): 0 4:1--4:25, 2013. doi:10.1145/2432622.2432626

    arXiv 2013

  12. [12]

    Concentration Inequalities: A Nonasymptotic Theory of Independence

    St\' e phane Boucheron, G\' a bor Lugosi, and Pascal Massart. Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford University Press, 2013

    2013

  13. [13]

    Nonparametric homogeneity tests

    Cristina Butucea and Karine Tribouley. Nonparametric homogeneity tests. Journal of Statistical Planning and Inference, 136 0 (3): 0 597--639, 2006. doi:10.1016/j.jspi.2004.08.003

    work page doi:10.1016/j.jspi.2004.08.003 2006

  14. [14]

    Priv'IT : Private and sample efficient identity testing

    Bryan Cai, Constantinos Daskalakis, and Gautam Kamath. Priv'IT : Private and sample efficient identity testing. In Proceedings of the 34th International Conference on Machine Learning ( ICML ) , volume 70 of Proceedings of Machine Learning Research, pages 635--644. PMLR, 2017

    2017

  15. [15]

    Tony Cai, Abhinav Chakraborty, and Lasse Vuursteen

    T. Tony Cai, Abhinav Chakraborty, and Lasse Vuursteen. Federated nonparametric hypothesis testing with differential privacy constraints: Optimal rates and adaptive tests. arXiv preprint arXiv:2406.06749, 2024

    Pith/arXiv arXiv 2024

  16. [16]

    Canonne and Yucheng Sun

    Cl\' e ment L. Canonne and Yucheng Sun. Optimal closeness testing of discrete distributions made (complex) simple. arXiv preprint arXiv:2204.12640, 2022

    Pith/arXiv arXiv 2022

  17. [17]

    Canonne, Gautam Kamath, Audra McMillan, Adam Smith, and Jonathan Ullman

    Cl\' e ment L. Canonne, Gautam Kamath, Audra McMillan, Adam Smith, and Jonathan Ullman. The structure of optimal private tests for simple hypotheses. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing ( STOC ) , pages 310--321. ACM, 2019. doi:10.1145/3313276.3316336

    arXiv 2019

  18. [18]

    Canonne, Gautam Kamath, Audra McMillan, Jonathan Ullman, and Lydia Zakynthinou

    Cl\' e ment L. Canonne, Gautam Kamath, Audra McMillan, Jonathan Ullman, and Lydia Zakynthinou. Private identity testing for high-dimensional distributions. In Advances in Neural Information Processing Systems 33 ( NeurIPS 2020) , pages 10099--10111, 2020

    2020

  19. [19]

    Optimal algorithms for testing closeness of discrete distributions

    Siu-On Chan, Ilias Diakonikolas, Gregory Valiant, and Paul Valiant. Optimal algorithms for testing closeness of discrete distributions. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms ( SODA ) , pages 1193--1203. SIAM, 2014. doi:10.1137/1.9781611973402.88

    doi:10.1137/1.9781611973402.88 2014

  20. [20]

    Distributed differential privacy via shuffling

    Albert Cheu, Adam Smith, Jonathan Ullman, David Zeber, and Maxim Zhilyaev. Distributed differential privacy via shuffling. In Advances in Cryptology -- EUROCRYPT 2019 , volume 11476 of Lecture Notes in Computer Science, pages 375--403. Springer International Publishing, 2019. doi:10.1007/978-3-030-17653-2_13

    work page doi:10.1007/978-3-030-17653-2_13 2019

  21. [21]

    Local goodness-of-fit testing for H\" o lder -continuous densities: Minimax rates

    Julien Chhor and Alexandra Carpentier. Local goodness-of-fit testing for H\" o lder -continuous densities: Minimax rates. Bernoulli, 31 0 (4): 0 2747--2771, 2025. doi:10.3150/24-BEJ1824

    work page doi:10.3150/24-bej1824 2025

  22. [22]

    Differentially private nonparametric hypothesis testing

    Simon Couch, Zeki Kazan, Kaiyan Shi, Andrew Bray, and Adam Groce. Differentially private nonparametric hypothesis testing. In Proceedings of the 2019 ACM SIGSAC Conference on Computer and Communications Security ( CCS ) , pages 737--751. ACM, 2019. doi:10.1145/3319535.3339821

    arXiv 2019

  23. [23]

    Kane, John Peebles, and Eric Price

    Ilias Diakonikolas, Themis Gouleakis, Daniel M. Kane, John Peebles, and Eric Price. Optimal testing of discrete distributions with high probability. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing ( STOC ) , pages 542--555. ACM, 2021. doi:10.1145/3406325.3450997

    arXiv 2021

  24. [24]

    Berrett, and Cristina Butucea

    Amandine Dubois, Thomas B. Berrett, and Cristina Butucea. Goodness-of-fit testing for H\" o lder continuous densities under local differential privacy. In Foundations of Modern Statistics, volume 425 of Springer Proceedings in Mathematics & Statistics, pages 53--119. Springer, 2023. doi:10.1007/978-3-031-30114-8_2

    work page doi:10.1007/978-3-031-30114-8_2 2023

  25. [25]

    The algorithmic foundations of differential privacy

    Cynthia Dwork and Aaron Roth. The algorithmic foundations of differential privacy. Foundations and Trends in Theoretical Computer Science, 9 0 (3--4): 0 211--407, 2014. doi:10.1561/0400000042

    doi:10.1561/0400000042 2014

  26. [26]

    Calibrating noise to sensitivity in private data analysis

    Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith. Calibrating noise to sensitivity in private data analysis. In Theory of Cryptography Conference ( TCC ) , pages 265--284. Springer, 2006. doi:10.1007/11681878_14

    doi:10.1007/11681878_14 2006

  27. [27]

    Amplification by shuffling: From local to central differential privacy via anonymity

    \'U lfar Erlingsson, Vitaly Feldman, Ilya Mironov, Ananth Raghunathan, Kunal Talwar, and Abhradeep Thakurta. Amplification by shuffling: From local to central differential privacy via anonymity. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms , pages 2468--2479. SIAM , 2019. doi:10.1137/1.9781611975482.151

    doi:10.1137/1.9781611975482.151 2019

  28. [28]

    Adaptive goodness-of-fit tests in a density model

    Magalie Fromont and B \'e atrice Laurent. Adaptive goodness-of-fit tests in a density model. The Annals of Statistics, 34 0 (2): 0 680--720, 2006. doi:10.1214/009053606000000119

    work page doi:10.1214/009053606000000119 2006

  29. [29]

    Differentially private chi-squared hypothesis testing: Goodness of fit and independence testing

    Marco Gaboardi, Hyun Lim, Ryan Rogers, and Salil Vadhan. Differentially private chi-squared hypothesis testing: Goodness of fit and independence testing. In Proceedings of the 33rd International Conference on Machine Learning ( ICML 2016) , volume 48 of Proceedings of Machine Learning Research, pages 2111--2120. PMLR, 2016

    2016

  30. [30]

    Borgwardt, Malte J

    Arthur Gretton, Karsten M. Borgwardt, Malte J. Rasch, Bernhard Sch \"o lkopf, and Alexander Smola. A kernel two-sample test. Journal of Machine Learning Research, 13 0 (25): 0 723--773, 2012

    2012

  31. [31]

    Yu. I. Ingster. Minimax testing of nonparametric hypotheses on a distribution density in the L_p metrics. Theory of Probability & Its Applications, 31 0 (2): 0 333--337, 1987. doi:10.1137/1131042

    work page doi:10.1137/1131042 1987

  32. [32]

    Yu. I. Ingster. Adaptive chi-square tests. Journal of Mathematical Sciences, 99 0 (2): 0 1110--1119, 2000. doi:10.1007/BF02673632

    work page doi:10.1007/bf02673632 2000

  33. [33]

    Yu. I. Ingster and Irina A. Suslina. Nonparametric Goodness-of-Fit Testing Under Gaussian Models, volume 169 of Lecture Notes in Statistics. Springer, New York, NY, 2003

    2003

  34. [34]

    Zeki Kazan, Kaiyan Shi, Adam Groce, and Andrew P. Bray. The test of tests: A framework for differentially private hypothesis testing. In Proceedings of the 40th International Conference on Machine Learning ( ICML ) , volume 202 of Proceedings of Machine Learning Research, pages 16131--16151. PMLR, 2023

    2023

  35. [35]

    Berrett, and Yi Yu

    Alexander Kent, Thomas B. Berrett, and Yi Yu. Locally differentially private two-sample testing. Biometrika, 2026. doi:10.1093/biomet/asag034. Advance article, asag034

    work page doi:10.1093/biomet/asag034 2026

  36. [36]

    Differentially private permutation tests

    Ilmun Kim and Antonin Schrab. Differentially private permutation tests. Journal of the American Statistical Association, pages 1--13, 2026. doi:10.1080/01621459.2025.2610033

    arXiv 2026

  37. [37]

    Minimax optimality of permutation tests

    Ilmun Kim, Sivaraman Balakrishnan, and Larry Wasserman. Minimax optimality of permutation tests. The Annals of Statistics, 50 0 (1): 0 225--251, 2022. doi:10.1214/21-AOS2103

    work page doi:10.1214/21-aos2103 2022

  38. [38]

    Human decisions and machine predictions

    Jon Kleinberg, Himabindu Lakkaraju, Jure Leskovec, Jens Ludwig, and Sendhil Mullainathan. Human decisions and machine predictions. The Quarterly Journal of Economics, 133 0 (1): 0 237--293, 2018. doi:10.1093/qje/qjx032

    doi:10.1093/qje/qjx032 2018

  39. [39]

    Earnshaw, Imran S

    Pang Wei Koh, Shiori Sagawa, Henrik Marklund, Sang Michael Xie, Marvin Zhang, Akshay Balsubramani, Weihua Hu, Michihiro Yasunaga, Richard Lanas Phillips, Irena Gao, Tony Lee, Etienne David, Ian Stavness, Wei Guo, Berton A. Earnshaw, Imran S. Haque, Sara Beery, Jure Leskovec, Anshul Kundaje, Emma Pierson, Sergey Levine, Chelsea Finn, and Percy Liang. WILDS...

    2021

  40. [40]

    Differentially private goodness-of-fit tests for continuous variables

    Seung Woo Kwak, Jeongyoun Ahn, Jaewoo Lee, and Cheolwoo Park. Differentially private goodness-of-fit tests for continuous variables. Econometrics and Statistics, 31: 0 81--99, 2024. doi:10.1016/j.ecosta.2021.09.007

    work page doi:10.1016/j.ecosta.2021.09.007 2024

  41. [41]

    On the statistical complexity of estimation and testing under privacy constraints

    Cl\' e ment Lalanne, Aur\' e lien Garivier, and R\' e mi Gribonval. On the statistical complexity of estimation and testing under privacy constraints. Transactions on Machine Learning Research, 2023 a

    2023

  42. [42]

    About the cost of central privacy in density estimation

    Cl\' e ment Lalanne, Aur\' e lien Garivier, and R\' e mi Gribonval. About the cost of central privacy in density estimation. Transactions on Machine Learning Research, 2023 b

    2023

  43. [43]

    Minimax optimal goodness-of-fit testing for densities and multinomials under a local differential privacy constraint

    Joseph Lam-Weil, B\' e atrice Laurent, and Jean-Michel Loubes. Minimax optimal goodness-of-fit testing for densities and multinomials under a local differential privacy constraint. Bernoulli, 28 0 (1): 0 579--600, 2022. doi:10.3150/21-BEJ1358

    work page doi:10.3150/21-bej1358 2022

  44. [44]

    Minimax optimal two-sample testing under local differential privacy

    Jongmin Mun, Seungwoo Kwak, and Ilmun Kim. Minimax optimal two-sample testing under local differential privacy. Journal of Machine Learning Research, 26 0 (252): 0 1--79, 2025

    2025

  45. [45]

    Private high-dimensional hypothesis testing

    Shyam Narayanan. Private high-dimensional hypothesis testing. In Proceedings of the 35th Conference on Learning Theory ( COLT ) , volume 178 of Proceedings of Machine Learning Research, pages 3979--4027. PMLR, 2022

    2022

  46. [46]

    Dissecting racial bias in an algorithm used to manage the health of populations

    Ziad Obermeyer, Brian Powers, Christine Vogeli, and Sendhil Mullainathan. Dissecting racial bias in an algorithm used to manage the health of populations. Science, 366 0 (6464): 0 447--453, 2019. doi:10.1126/science.aax2342

    doi:10.1126/science.aax2342 2019

  47. [47]

    Differentially private equivalence testing for continuous distributions and applications

    Daniel Omer and Or Sheffet. Differentially private equivalence testing for continuous distributions and applications. In Advances in Neural Information Processing Systems 37 ( NeurIPS 2024) , pages 2507--2527, 2024. doi:10.52202/079017-0082

    work page doi:10.52202/079017-0082 2024

  48. [48]

    Lawrence, editors

    Joaquin Qui \ n onero-Candela, Masashi Sugiyama, Anton Schwaighofer, and Neil D. Lawrence, editors. Dataset Shift in Machine Learning. MIT Press, 2009

    2009

  49. [49]

    A differentially private kernel two-sample test

    Anant Raj, Ho Chung Leon Law, Dino Sejdinovic, and Mijung Park. A differentially private kernel two-sample test. In Machine Learning and Knowledge Discovery in Databases ( ECML-PKDD ) , volume 11906 of Lecture Notes in Computer Science, pages 697--724. Springer, 2020. doi:10.1007/978-3-030-46150-8_41

    work page doi:10.1007/978-3-030-46150-8_41 2020

  50. [50]

    A new class of private chi-square hypothesis tests

    Ryan Rogers and Daniel Kifer. A new class of private chi-square hypothesis tests. In Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, volume 54 of Proceedings of Machine Learning Research, pages 991--1000. PMLR, 2017

    2017

  51. [51]

    Aggregation of statistical evidence under exchangeability

    Antonin Schrab, Rajen Shah, Arthur Gretton, and Ilmun Kim. Aggregation of statistical evidence under exchangeability. arXiv preprint arXiv:2607.15823, 2026

    Pith/arXiv arXiv 2026

  52. [52]

    Spokoiny

    Vladimir G. Spokoiny. Adaptive hypothesis testing using wavelets. The Annals of Statistics, 24 0 (6): 0 2477--2498, 1996. doi:10.1214/aos/1032181163

    arXiv 1996

  53. [53]

    A statistical framework for differential privacy

    Larry Wasserman and Shuheng Zhou. A statistical framework for differential privacy. Journal of the American Statistical Association, 105 0 (489): 0 375--389, 2010. doi:10.1198/jasa.2009.tm08651

    doi:10.1198/jasa.2009.tm08651 2010

  54. [54]

    Statistical Inference in the Differential Privacy Model

    Huanyu Zhang. Statistical Inference in the Differential Privacy Model. Ph.D. dissertation, Cornell University, August 2021

    2021

This paper was first reviewed by deepseek-v4-flash on July 31, 2026.