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Unifying Privacy Accounting: Information Equivalence and Information Loss

Buxin Su, Qiaoshi Yang, Yiding Su, Chendi Wang

arXiv:2610.02414v1cs.CRcs.ITmath.ITmath.STstat.TH

TL;DR

Privacy guarantees use several curve-based notions, raising the question of whether choosing among them preserves the exact privacy information of a fixed pair and what is lost by compressing RDP into zCDP. The paper proves an information-equivalence result and quantifies the RDP–zCDP gap. The gap is zero for Gaussian noise but can grow under other mechanisms and independent composition.

  • Problem

    A single (ε, δ) pair gives only one point on a privacy profile and may not order mechanisms, motivating the question of whether full privacy notions preserve the exact information of a fixed pair.

    Source: §1 · p. 2

  • Method

    The paper establishes information equivalence among full curve-based notions for a fixed pair and measures RDP compression loss using the gap between the exact RDP curve and its zCDP envelope.

    Source: §1 · p. 3, §3 · p. 7

  • Results

    For a fixed pair, directional profiles, trade-off functions, PLD, and conditionally exact RDP are information-equivalent; taking profile maxima or compressing RDP to zCDP can lose information.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The paper states that information loss can require additional noise and reduce accuracy, while retaining full RDP can reduce calibration noise.

    Source: §1 · p. 3

    The equivalence theorem concerns one fixed pair, not a mechanism-level RDP guarantee formed by taking a potentially different maximizing pair at each order.

    Source: §2 · p. 6

Abstract

from arXiv · show

Differential privacy (DP) admits several notions, but the choice among them may affect both privacy analysis and utility. In this paper, we consider four mainstream curve-based privacy notions within a unified information-theoretic framework. For a fixed ordered pair of output distributions, we establish information equivalence among the two directional privacy profiles of $(\varepsilon,\delta)$-DP, the pair of hypothesis-testing trade-off functions, and the extended privacy-loss distribution. The exact R\'enyi differential privacy (RDP) curve joins this equivalence class whenever it is finite at some order greater than one. Under this mild condition, choosing among these notions changes only their semantic interpretation and computational requirements. In contrast, taking the maximum of the directional privacy profiles or compressing the RDP curve into a single zero-concentrated differential privacy (zCDP) parameter can lose information. We quantify the information loss between the exact RDP curve and its zCDP bound for standard noise mechanisms. This gap is zero for Gaussian noise but generally positive for Gaussian-mixture, Laplace, discrete Gaussian, and Poisson-subsampled Gaussian mechanisms. Moreover, this gap grows linearly with the number of independently composed mechanisms. Our information-theoretic perspective has practical consequences. At the same certified privacy level, retaining the full RDP curve rather than using zCDP reduces the required noise variance by up to $45\%$ for Gaussian-mixture noise in workloads comparable in size to the American Community Survey. For DP-SGD on Fashion-MNIST under Poisson subsampling, an RDP-based privacy accountant improves test accuracy by up to $8.73$ percentage points compared to a zCDP-based accountant when both are calibrated to the same $(\varepsilon,\delta)$ guarantee.

1 Introduction

The introduction asks whether full privacy notions retain the same information for a fixed pair of output distributions, and how much information zCDP loses by compressing an RDP curve. It presents equivalence among directional privacy curves and related notions, then examines the resulting accounting trade-offs.

1 Introduction

  • The motivating questions are whether full notions preserve the exact privacy information of a fixed pair and how much information a single zCDP parameter discards.A single (ε, δ) pair is only one operating point and may not totally order mechanisms.Source: §1 · p. 2
  • Earlier Census re-accounting reduced a discrete-Gaussian variance proxy by 15.08%–24.82% at nearly the same certified privacy, while Figure 2 describes a Fashion-MNIST RDP-versus-zCDP comparison at matched privacy.The Fashion-MNIST figure reports test accuracy and cross-entropy across training-step settings and training histories.Source: §1 · p. 3, Figure 2 · p. 4

Our Contributions

  • For a fixed ordered pair, the two one-sided DP profiles, trade-off functions, and extended PLD contain the same information as the exact RDP curve under a mild condition.Taking the maximum of the one-sided profiles can lose information.Source: §1 · p. 3
  • The zCDP–RDP gap is zero for Gaussian noise, scales as s^2 for Gaussian-mixture noise and s^4 for Laplace noise, and grows under Poisson subsampling as q decreases.The gap adds under independent composition; as q approaches zero, it approaches the entire zCDP bound.Source: §1 · p. 3
  • The paper’s results support retaining the full RDP curve or an information-equivalent notion for tighter accounting, and using zCDP when the gap is negligible.The introduction states that information loss can require additional noise and reduce accuracy.Source: §1 · p. 3

1 Introduction

  • The introduction frames the work against prior connections among privacy curves, hypothesis testing, privacy-loss distributions, and RDP.It identifies the paper’s novelty as quantifying order-wise RDP-to-zCDP information loss and its local expansion and accumulation under independent composition.Source: §1 · pp. 3–4

2 Information Equivalence between Exact Privacy Notions

For a fixed pair of output distributions, the paper establishes equivalence among directional DP curves, hypothesis-testing trade-off functions, and the extended privacy-loss distribution. The exact RDP curve also determines these notions when it is finite at one order greater than one.

2 Information Equivalence between Exact Privacy Notions

  • The one-sided DP curve is the smallest δ satisfying P(A) ≤ e^εQ(A) + δ for every event A; exchanging P and Q defines the reverse curve.The paper defines the trade-off function as the smallest type II error among tests with type I error at most a.Source: §2 · p. 4
  • The pair of one-sided (ε, δ)-DP curves, trade-off functions, and extended PLD determine one another; exact RDP joins this equivalence when finite at some order greater than one.The condition holds for the mechanisms studied here, but not when every positive exponential moment of privacy loss is infinite.Source: §2 · p. 5
  • Finite mean privacy loss alone is insufficient for recovery: the examples have different forward curves despite finite means, while their RDP values are infinite at every order above one.The example also reports differing forward-curve values at ε = 1.Source: §2 · p. 5
  • Two pairs can share the same two-sided DP curve yet have different one-sided curves, trade-off functions, PLDs, and RDP curves.In the example, the reverse curves differ at log 3, and exp(D2) is 265/88 for one pair versus 3 for the other.Source: §2 · p. 6
  • For additive noise symmetric about zero, the two one-sided curves coincide, so the usual two-sided curve retains all the information.The paper gives Gaussian and Laplace noise as examples.Source: §2 · p. 6
  • The equivalence concerns one fixed pair, not a mechanism-level RDP guarantee whose maximizing neighboring pair may vary with order.A conversion from a mechanism-level bound must hold for every pair consistent with it.Source: §2 · p. 6

3 The Information Gap

The information gap measures what zCDP discards by compressing the exact RDP curve into a single parameter. It is zero for Gaussian noise, can be expanded locally for small query changes, quantifies privacy-bound differences, and adds under independent composition.

3 The Information Gap

  • For Gaussian noise, Γ(t) is zero at every order, while the gap measures the distance between the exact RDP curve and its zCDP envelope.Γ(t)/t is the vertical distance; multiplying by t puts Γ on the scale of the privacy-loss cumulant-generating function, which adds under independent composition.Source: §3 · p. 7

3.1 Local Expansion

  • For small query changes, the gap’s local expansion is determined by the mean privacy loss and the third and fourth cumulants of privacy loss.The terms are generally of orders s², s³, and s⁴; all three vanish for Gaussian noise.Source: §3.1 · p. 8

3.2 From the Gap to Privacy Bounds

  • Near the order optimizing the privacy bound, Γ(t)/t bounds the cost of zCDP compression; identifying a useful interval requires mechanism-specific analysis.If an interval containing the exact-RDP optimizing order has Γ(t)/t at most η throughout, then the zCDP ε bound exceeds the exact-RDP ε bound by at most η.Source: §3.2 · pp. 8–9

3.3 Information Loss under Independent Composition

  • For n independent copies of a fixed pair, the product pair’s information gap is nΓ(t), so the gap grows linearly with the number of copies.For nonidentical components, the product pair’s optimal zCDP parameter may be smaller than the sum of component parameters; the composition gap has two nonnegative terms.Source: §3.3 · p. 10
  • The local expansions of independent component families add componentwise under the assumptions of Theorem 3.1 for each component.Source: §3.3 · p. 10

4 Information Gaps for Common Mechanisms

Across common mechanisms, the zCDP–RDP information gap ranges from zero to positive and depends on the noise family, subsampling rate, allocation, tail parameter, and RDP order. The section quantifies these differences for continuous, subsampled, and lattice mechanisms.

4.1 Continuous Additive Noise Mechanisms

  • Gaussian noise has zero gap, while Gaussian-mixture and Laplace gaps scale as s^2 and s^4, respectively, for small shifts.Figure 4(a) reports fitted log–log slopes of 2.00 and 4.00 for the mixture and Laplace gaps.Source: §4.1 · pp. 10–11, Figure 4 · p. 11

4.2 Poisson Subsampling

  • As Poisson subsampling decreases, the gap rises; as q tends to zero, it approaches the full zCDP bound, although the exact RDP curve tends to zero.At q = 1, there is no subsampling and the Gaussian step has zero gap.Source: §4.2 · pp. 11–12

4.3 Allocation under a Fixed zCDP Composition Parameter

  • At a fixed total zCDP parameter, allocation changes the composed gap: it rises from 0.037202015 to 0.193535993 for Laplace noise, but falls from 0.435179700 to 0.342190807 for Gaussian-mixture noise when moving from uniform to concentrated.The Laplace result follows from strict convexity; the Gaussian-mixture comparison is numerical evidence on the displayed grid.Source: §4.3 · p. 13, Figure 5 · p. 14

4.4 Tail Parameters and the Information Gap

  • For Student-t noise, the small-shift quartic gap coefficient peaks near ν = 6.013 and tends to zero in the Gaussian limit, so lighter tails do not monotonically order the gap.For generalized Gaussian noise, the displayed gap decreases to zero at β = 2; the fixed-shift curves are numerical, and their dashed local approximations are not asserted to bound the gaps.Source: §4.4 · pp. 14–16, Figure 6 · p. 15

4.5 Discrete and Lattice Mechanisms

  • Discrete Gaussian gaps are periodic in the RDP order offset, vanishing exactly when (1 + t)r is an integer, while discrete Laplace gaps are positive for t > 0.For discrete Gaussian noise, the gap is 1/r-periodic and exponentially small as σ grows.Source: §4.5 · p. 16, Figure 7 · p. 17

5 Experiments

At the same certified privacy level, the experiments compare noise and utility under RDP and zCDP calibration. RDP saves noise for Gaussian-mixture workloads and improves DP-SGD accuracy most at the lowest sampling rate, while the methods converge at higher rates.

5.1 ACS-sized Workload

  • For each of 20 states, the ACS workload uses 330–1060 median-earnings queries and compares the smallest per-query variance certifying (ε0, δ0) = (3, 10−5).Noise is centered and independent across queries, so per-query MSE equals variance; this is an illustration on published aggregates, not a person-level DP mechanism.Source: §5.1 · p. 17
  • 0.552–0.557 of the zCDP variance: retaining the RDP curve requires about 45% lower noise variance for Gaussian-mixture noise; Gaussian calibrations coincide and Laplace ratios are 0.989–0.997.The Gaussian-mixture ∆MSE is 652.3–2115.7 and the Laplace ∆MSE is 16.170–16.205 in units of ($1000)2 per query.Source: Figure 8 · p. 18, §5.1 · p. 18

5.2 DP-SGD

  • 8.73 ± 6.10 percentage points: at q = 0.005, RDP calibration gains paired test accuracy over zCDP; at q ≥0.02, all three methods agree within seed-to-seed variability.Across the q sweep, RDP test accuracy stays near 65.2%; the zCDP penalty shrinks as q grows.Source: Figure 10 · p. 19, §5.2 · p. 19
  • The DP-SGD experiments train a two-convolutional-layer Fashion-MNIST CNN with Poisson sampling, clipping C = 1, and noise calibrated separately for each q and step count.Each setting uses five seeds, with paired initialization, minibatches, and noise draws for the RDP and zCDP runs.Source: §5.2 · p. 19
  • At fixed n = 500, changing q also changes batch size and epoch count (nq from 2.5 to 25), so the observed q trend is consistent with, but not a direct consequence of, Proposition 4.2.Figure 10 reports final accuracy and cross-entropy across q and training histories at q = 0.005.Source: Figure 10 · p. 19, §5.2 · p. 20
  • As composition count increases, RDP accuracy rises from 61.866% at n = 250 to 70.522% at n = 2000, while zCDP accuracy peaks at 66.354% at n = 1000 and falls to 64.600% at n = 2000.The paired test-loss difference grows with n, but the paired accuracy trend and endpoint contrast are not significant; cross-entropy is the cleaner measure.Source: §5.2 · p. 20

6 Discussion and Limitations

The discussion distinguishes information-preserving privacy representations from lossy compressions and qualifies the theorem’s scope. The proof recovers privacy-loss and trade-off representations from the directional curves, and recovers them from exact RDP under a finite-order condition.

6 Discussion and Limitations

  • The two directional (ε, δ)-DP curves, trade-off functions, PLD, and exact RDP curve carry the same information for a fixed pair when RDP is finite at one order.The usual two-sided curve discards directional information, whereas zCDP can lose information even for a fixed pair; zCDP is lossless for Gaussian noise without subsampling.Source: §6 · p. 20
  • The equivalence theorem concerns one fixed distribution pair, not a mechanism-level RDP curve whose maximizing neighboring pair may vary with order.The discussion also assumes independent components for composition, and local expansions require small shifts and bounded orders.Source: §6 · p. 20
  • The experimental evidence covers three noise families, ACS-sized workloads, and one Fashion-MNIST model with five seeds; broader datasets, optimizers, and deployments remain future work.The discussion also identifies mechanism-level guarantees, adaptive composition, and rigorous numerical evaluation of some optimizing orders as future directions.Source: §6 · p. 20

A.1 Proof of Theorem 2.1

  • The proof recovers the privacy-loss law from the exact RDP curve when it is finite at some order greater than one, then uses it to recover both directional profiles and trade-off functions.It tilts the privacy-loss law, uses moment-generating-function uniqueness, and untilts to obtain the original law.Source: §A.1 · p. 25

A.2 Proof of Corollary A.1

The appendix proves the symmetry result for centrally symmetric additive noise and supplies examples showing what privacy summaries can fail to distinguish. It also establishes a Chernoff step used in the conversion-bound proof.

A.2 Proof of Corollary A.1

  • For centrally symmetric additive noise, reflection exchanges the neighboring output distributions, making the directional profiles, trade-off functions, and RDP values coincide in both directions.The common one-sided curve, exact f-DP curve, and privacy-loss law determine one another; finite exact RDP at an order above one also recovers the other notions.Source: §A.2 · p. 26

A.3 Proof of Example 2.2

  • Example 2.2 has an everywhere-infinite RDP curve for w > 2, yet its forward profiles differ between w = 3 and w = 4.The contrast shows that without a finite RDP order, the converse recovery condition need not apply.Source: §A.3 · p. 26

A.4 Proof of Example 2.3

  • All curves vanish after their stated cutoffs, so the pointwise maxima agree; nevertheless, the RDP values differ: 265/88 for pair A and 3 for pair B.The strongest symmetric f-DP guarantee also agrees for the two pairs.Source: §A.4 · p. 27

B.1 Proof of Lemma B.1

  • The hockey-stick Chernoff proof maximizes y^t(1 − y) at y = t/(1 + t), yielding the inequality used in the lemma.For x ≥ 0 the left side is zero; for x < 0 the substitution y = e^x puts y in (0, 1).Source: §B.1 · p. 27

B.2 Proof of the conversion bounds in Section 3.2

  • The conversion proof applies the Chernoff bound to privacy loss and derives a fixed-order conversion; under a zCDP bound, optimization yields the conversion in (3.8) with a unique minimizer.Dropping the prefactor recovers the original conversion of Mironov (2017).Source: §B.2 · pp. 27–28

B.3 Proof of Proposition 3.2

These proofs establish bounds and expansions for the zCDP–RDP gap, then apply them to composition and noise mechanisms. Their scope includes endpoint-control assumptions, independent composition, and numerical rather than proved concavity evidence for a Gaussian-mixture comparison.

B.3 Proof of Proposition 3.2

  • The composition bound closes using Γ ≥ 0 and Γ(tR)/tR ≤ η when tR lies in the interval I.Source: §B.3 · p. 28

B.4 Proof of Lemma B.2

  • The fourth-order expansion lemma applies to continuous and discrete mechanisms under its smoothness and boundedness conditions and retains an explicit fifth-order remainder.The local expansion theorem follows by applying the lemma with κ1(Ls) = E_Ps Ls.Source: §B.4 · p. 28, §B.5 · p. 29

B.6 Proof of Theorem 3.3

  • For n independent copies of one pair, both the optimal zCDP parameter and the gap Γ scale by n.The proof uses additivity of log moment-generating functions under independence.Source: §B.6 · p. 29

C.1 Proof of Lemma C.1

  • Under the endpoint-localization assumptions, the gap supremum is attained at the zero-order endpoint, giving ρ(s) = E_Ps Ls and As = 0.The localization proof uses the factorization K(t) = r²t(t + 1)Hs(rt).Source: §C.1 · p. 30

C.2 Derivation sketch for (C.6) and (C.7)

  • The smooth-location derivation obtains the fourth-order coefficient through integration by parts and notes that a kink in the log-density can change that coefficient.Source: §C.2 · p. 31

C.3 Proof of Proposition 4.1

  • Gaussian noise has Γ(t) = 0 exactly because its CGF is quadratic and equals the zCDP envelope.The proof also gives ρ(s) = s²/2 for Gaussian noise.Source: §C.3 · p. 31

C.4 Proof of Proposition 4.2

  • For Poisson subsampling, the one-step gap decreases strictly as q increases and reaches zero at q = 1; its limiting behavior is evaluated at q = 0.The proof identifies the one-step optimal zCDP parameter as 1/(2z²) for every q > 0.Source: §C.4 · pp. 33–34

C.5 Proof of Proposition 4.3

  • The composition comparison uses Jensen and chord inequalities; Gaussian-mixture Φ1 concavity is supported numerically on the plotted interval, not proved.For the displayed grid, changing Hermite resolution and using 501 grid points retained negative interior centered second differences.Source: §C.5 · p. 34

C.6 Proof of Proposition 4.4

The proof controls the Student-t privacy-loss CGF and its small-shift expansion to characterize the zCDP–RDP gap. The quartic coefficient has a unique maximizer near ν = 6.013.

C.6 Proof of Proposition 4.4

  • The proof uses the bounded Student-t score and a beta-distributed transform to control the CGF uniformly over orders.The transform is Yν = Z²/(ν + Z²) ∼ Beta(1/2, ν/2); moment comparison and tail control establish the quantitative sub-Gaussian condition.Source: §C.6 · pp. 34–35
  • The normalized CGF has a finite endpoint at t = 0, and the proof concludes that ρν(s) is finite for every real shift s.The density’s evenness makes the CGF and ρν(s) even functions of s.Source: §C.6 · p. 34
  • Symmetry eliminates the cubic term, and the small-shift gap has a quartic leading term with an O(|s|^6) remainder.The fifth-order Taylor coefficient vanishes because both the CGF and the mean privacy loss are even in s.Source: §C.6 · p. 35
  • The quartic coefficient’s unique maximizer on [1, ∞) is ν ≈ 6.013052641.The proof establishes uniqueness through the single sign change in the coefficients of p(ν).Source: §C.6 · p. 36

C.7 Proof of Proposition 4.5

These proofs derive local gap expansions for generalized-Gaussian and lattice noise, while the supplementary material checks the numerical comparisons and computational cutoffs used in the plots.

C.7 Proof of Proposition 4.5

  • For generalized-Gaussian noise, β = 1 gives χ1 = 1/12, while β = 2 gives χ2 = 0 and Γ(t) = 0 for every t > 0.The β = 1 cumulants satisfy κ3(Us) = −s + O(s²) and κ4(Us) = −2 + O(|s|); the local gap term is χβ[t(t + 1)]²s⁴ + o(|s|⁴).Source: §C.7 · pp. 37–38

C.8 Proof of Proposition 4.6

  • For discrete Gaussian noise, the zCDP envelope is exact at integer-aligned orders, with a periodic zero set in the order-dependent gap.The theta-sum term is nonpositive and vanishes when (1 + t)r is an integer.Source: §C.8 · p. 38

D Supplementary Figures and Numerical Details

  • The supplementary comparisons at t = 1.5 retain the reported local-expansion patterns, and numerical checks assess quadrature accuracy, order truncation, and noise calibration.The figures also repeat the experimental comparisons at (ε, δ) = (5, 10⁻⁵).Source: D Supplementary Figures and Numerical… · p. 39

D.1 Computational Setup

  • The computational setup uses quadrature for Gaussian-mixture, Student-t, and generalized-Gaussian expectations, and separately checks the large-order cutoff.Repeating the Gaussian-mixture calculation with 384 rather than 192 Hermite nodes preserves the reported interior concavity pattern numerically.Source: §D.1 · p. 39

D.2 Results at an Alternative Reporting Order

At the alternative order t = 1.5, the mechanism-gap patterns reported at t = 1 remain visible. The Poisson and allocation comparisons also retain their stated directions.

D.2 Results at an Alternative Reporting Order

  • The displayed comparisons repeat the fixed-order analyses at Rényi order α = 2.5 with mechanism parameters and numerical grids unchanged.The lattice examples need no duplicate fixed-order panel because Figure 7 already spans 0 ≤ t ≤ 3.Source: §D.2 · p. 39
  • The Gaussian gap remains zero, while fitted slopes of 2.000 for Gaussian-mixture and 3.997 for Laplace agree with quadratic and quartic expansions.The Student-t curve remains nonmonotone, and the generalized-Gaussian curve reaches zero at β = 2.Source: §D.2 · p. 40, Figure 11 · p. 40
  • At t = 1.5, Poisson gaps still decrease from positive-q right limits to zero; concentrating allocation increases the Laplace gap and decreases the computed Gaussian-mixture gap.The composition comparison uses n = 20 and ρsum = 5; the allocation comparison uses n = 8 and ρsum = 1/2.Source: Figure 12 · p. 41

D.3.1 Experimental Setup and Metadata

The ACS study compares RDP- and zCDP-calibrated noise on 20 state-level workloads, then repeats the comparison at a weaker privacy target. At that target, Gaussian calibrations still coincide, while mixture MSE differences exceed Laplace differences.

D.3.1 Experimental Setup and Metadata

  • The ACS workloads use the top 20 states by county count; each state’s mi is its complete-county count and di = 5mi is its number of county-by-education queries.The study excludes 137 incomplete rows from the selected states without imputation and compares calibrated scales using 20,000 paired draws with shared base noise.Source: §D.3.1 · p. 42, Table 1 · p. 42

D.3.2 Results at an Alternative Privacy Target

  • At (ε, δ) = (5, 10⁻⁵), Gaussian calibrations still coincide, Gaussian-mixture MSE differences still exceed Laplace differences, and absolute differences are smaller.The alternative-target figures repeat the comparisons across the same 20 workloads.Source: §D.3.2 · p. 42, Figure 13 · p. 43, Figure 14 · p. 43

D.4.1 Experimental Setup and Metadata

The Fashion-MNIST DP-SGD experiment uses controlled sweeps over sampling rates and composition counts, with paired methods, seeds, and random streams. The section also documents endpoint reporting and numerical calibration checks.

D.4.1 Experimental Setup and Metadata

  • Fashion-MNIST uses 60,000 training and 10,000 test images, a two-stage convolutional network, flat clipping at C = 1, and Poisson-sampled Gaussian noise.The runs use SGD and a shared 2000-step cosine learning-rate schedule.Source: §D.4.1 · p. 44
  • The sampling-rate sweep tests five q values at n = 500, while the composition-count sweep tests five n values at q = 0.01, each with five paired seeds.Within each setting and seed, the three methods share initialization and Poisson sampling; RDP and zCDP also share their standard-normal noise stream.Source: §D.4.1 · p. 44
  • Test metrics and a deterministic training probe are recorded every 25 steps, and Table 3 reports nine settings at (ε, δ) = (3, 10−5).Mean RDP and clipping-only test accuracies differ by at most 0.08 percentage points; zCDP accuracy varies with calibrated noise.Source: §D.4.1 · p. 44, Table 3 · p. 45
  • All scheduled trajectories and endpoint results completed, with matching hashes confirming the paired initialization, sampling, and private-noise streams.Calibration errors in ε are at most 8.67 × 10−10 at ε = 3 and 1.52 × 10−9 at ε = 5; grid refinement and q = 1 checks are also reported.Source: §D.4.1 · p. 44

D.4.2 Results at an Alternative Privacy Target

At (ε, δ) = (5, 10−5), the RDP–zCDP test-loss gap remains smaller than at the stricter target and varies with sampling rate and composition count. Accuracy differences do not show a corresponding monotone pattern.

D.4.2 Results at an Alternative Privacy Target

  • The RDP–zCDP test-cross-entropy differences are smaller at the weaker privacy target, decrease toward zero as q increases, and grow with n.The sampling-rate and composition-count sweeps are shown in Figures 15 and 16.Source: §D.4.2 · p. 46, Figure 15 · p. 46, Figure 16 · p. 47
  • Accuracy differences are nonmonotone, with no significant trend across q (p = 0.106) or n (p = 0.361).Thus the test-loss pattern persists without a corresponding monotone accuracy pattern.Source: §D.4.2 · p. 46
  • 0.154 at q = 0.005 falls to approximately zero at q = 0.05, while the paired test-loss difference rises from 0.012 at n = 250 to 0.182 at n = 2000.These are paired means of ∆Ltest = LZ − LR; Table 4 reports the endpoint metrics at (ε, δ) = (5, 10−5).Source: §D.4.2 · p. 46, Table 4 · p. 48

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