Blanket Divergence in Differential Privacy

• Blanket divergence is a privacy metric that quantifies how shuffling local randomizer outputs amplifies privacy by leveraging the minimal 'blanket density' and the shuffle index.
• It provides a rigorous framework through asymptotic analysis and concentration bounds to measure privacy loss under varying participation rates.
• Practical implementations use FFT-based algorithms to compute divergence for mechanisms like k-randomized response and generalized Gaussian, ensuring controlled error in privacy accounts.

Blanket divergence is a fundamental measure in the analysis of privacy amplification by shuffling in distributed data collection, particularly within the shuffle model of local differential privacy. As established in the work of Takagi et al., the blanket divergence encapsulates the contribution of the data-independent part of the local randomizer (“blanket density”) to the privacy guarantee when output messages are shuffled before aggregation. Its asymptotic behavior is governed by a single parameter—the shuffle index $χ$—that quantifies the efficiency of privacy amplification with respect to both the randomizer and the participation rate.

1. Formal Definition of Blanket Divergence

Let $\mathcal R: \mathcal X \to \mathcal Y$ denote a local randomizer, such that for each $x\in\mathcal X$, the output distribution has density $\mathcal R_x(y)$ relative to a base measure on $\mathcal Y$. The blanket density is given by: $$\underline{\mathcal R}(y) = \inf_{x\in\mathcal X}\mathcal R_x(y) \qquad \gamma = \int_{\mathcal Y}\underline{\mathcal R}(y)\,dy \qquad \mathcal R_{\mathrm{BG}}(y) = \frac{1}{\gamma}\underline{\mathcal R}(y)$$ For neighboring inputs $x_1\ne x_1'$, and parameter $\epsilon\ge 0$, define the privacy-amplification random variable: $$l_\epsilon(y) = \frac{\mathcal R_{x_1}(y) - e^\epsilon \mathcal R_{x_1'}(y)}{\mathcal R_{\mathrm{BG}}(y)}, \quad y \sim \mathcal R_{\mathrm{BG}}$$ The blanket divergence is then the hockey-stick divergence bound: $$\mathcal D_{e^\epsilon, n, \mathcal R_{\mathrm{BG}},\gamma}^{\mathrm{blanket}}(\mathcal R_{x_1}\|\mathcal R_{x_1'}) = \frac{1}{n\gamma} \mathbb E\left[\max\left\{\sum_{i=1}^M l_\epsilon(Y_i),\, 0\right\}\right]$$ where $M \sim \mathrm{Binomial}(n, \gamma)$ and $Y_1,\dots,Y_M$ are i.i.d. samples from $\mathcal R_{\mathrm{BG}}$. Alternatively, after a size-biasing argument,

$$\mathcal D^{\mathrm{blanket}} = \mathbb E\left[l_\epsilon(Y_1)\left\{\Pr\left[\sum_{i=1}^{M'} l_\epsilon(Y_i) > 0\,|\,Y_1\right]\right\}\right]$$

with $M'\sim 1+\mathrm{Binomial}(n-1,\gamma)$. The blanket $\mathcal R_{\mathrm{BG}}$ represents the participation-weighted minimal output probability, and $l_\epsilon(Y_i)$ quantifies the per-sample privacy loss; the aggregate divergence measures the degree of privacy under random participation and maximal adversarial alignment.

2. Asymptotic Expansion Under Central Limit Regime

Under mild moment and nondegeneracy assumptions, without requiring pure-LDP, consider a vanishing privacy parameter $\epsilon_n \to 0$, subject to $\epsilon_n=\omega(n^{-1/2})$ and $\epsilon_n=O(\sqrt{\log n/n})$. Let

$$Z_i = B_i\,l_{\epsilon_n}(Y_i),\quad B_i \sim \mathrm{Bernoulli}(\gamma),\quad S_n = \sum_{i=1}^n Z_i$$

with $\mathrm{Var}(Z_i) = \sigma_{\epsilon_n}^2>0$ and mean $\mu_{\epsilon_n} = \gamma(1-e^{\epsilon_n})$. Setting

$$t_n = -\frac{n\mu_{\epsilon_n}}{\sigma_{\epsilon_n}\sqrt{n}}$$

the blanket divergence admits an asymptotic expansion (via Edgeworth and moderate deviation theory): $$\mathcal D^{\mathrm{blanket}}_{e^{\epsilon_n}, n, \mathcal R_{\mathrm{ref}}, \gamma} = \varphi(\chi \epsilon_n \sqrt{n}) \,\frac{1}{\chi^3 \epsilon_n^2 n^{3/2} (1+o(1))}$$ where $\varphi$ is the standard normal density and $\chi$ is the shuffle index. This result provides a precise quantification of privacy amplification, showing that the leading term depends only on $\chi$, establishing the universally amplified regime of $\epsilon_n$ via shuffling.

3. The Shuffle Index $χ$ and Mechanism Dependence

The shuffle index is defined as: $$\chi := \frac{\sqrt{\gamma}}{\sigma} \quad \sigma^2 := \mathrm{Var}\left(l_0(Y; x_1, x_1'; \mathcal R_{\mathrm{ref}})\right)$$ In upper bound analysis, $$\mathcal R_{\mathrm{ref}} = \mathcal R_{\mathrm{BG}}$$; in lower bound analysis, $\mathcal R_{\mathrm{ref}}=\mathcal R_x$ for some $x$.

Interpretation: $\chi$ characterizes the “shuffle efficiency,” quantifying how blanket mass and randomizer variability interact to yield the $\epsilon\to\epsilon\sqrt{n}$ privacy amplification regime. Higher $\chi$ yields stronger amplification.

Example Computations:

• For $k$-randomized response ($k$-RR) with local $\epsilon_0$:

$$p = \frac{e^{\epsilon_0}}{e^{\epsilon_0}+k-1}\,,\, q = \frac{1}{e^{\epsilon_0}+k-1}$$

$$\gamma = kq,\quad \sigma^2 = 2k(p-q)^2,\quad \chi_{\mathrm{lo}} = \sqrt{\frac{q}{2(p-q)^2}}$$

• For the generalized Gaussian mechanism:

$$\gamma = \int \inf_{x\in[0,1]} \frac{\beta}{2c\Gamma(1/\beta)} e^{-|y-x|^\beta/c^\beta} dy$$

and $\sigma^2$ is the variance of $l_0(y)$ (as above). Numerical or closed-form computation arises for $\beta=1,2$.

4. Tightness of Upper and Lower Bounds: Structural Conditions

Theorem 3.4 (Structural Condition) establishes the regime under which the blanket divergence bounds are tight. For all pairs $(x_1, x_1')$, define $$A(x_1,x_1') = \{y: \mathcal R_{x_1}(y) \ne \mathcal R_{x_1'}(y)\}$$, and shuffle indices $\chi_{\mathrm{lo}}, \chi_{\mathrm{up}}$ as the infima over references $\mathcal R_{\mathrm{BG}}, \mathcal R_x$, respectively.

• Always $\chi_{\mathrm{up}} \geq \chi_{\mathrm{lo}}$.
• Equality ($\chi_{\mathrm{up}} = \chi_{\mathrm{lo}}$) holds for a pair $(x_1^*, x_1'^*)$ iff there exists $x^* \in \mathcal X$ s.t.

$$\mathcal R_{x^*}(y) = \gamma\, \mathcal R_{\mathrm{BG}}(y) = \inf_{z\in\mathcal X} \mathcal R_z(y) \quad \text{for a.e. } y \in A(x_1^*, x_1'^*)$$

For $k$-RR with $k\ge3$, the minimal variance reference exactly saturates the blanket on the disagreement set, so the band collapses (asymptotically exact bounds). For generalized Gaussian mechanisms, no single $x$ saturates the blanket over the full disagreement region, resulting in distinct $\chi_{\mathrm{lo}}, \chi_{\mathrm{up}}$.

5. Asymptotic Privacy Band for $(\epsilon_n, \delta_n)$-DP

Fix a target $\delta_n \approx \alpha/n$, $\alpha>0$, and for any $\chi>0$, define: $$\epsilon_n(\alpha, \chi) = \ln\left( 1+\sqrt{\frac{2}{\chi^2 n} W\left( \frac{\sqrt{n}}{2\alpha\chi\sqrt{2\pi}} \right) }\right)$$ where $W$ is the principal Lambert-$W$ function.

There exist constants

$$\underline\chi_{\mathrm{up}} = \inf_{x_1\ne x_1'}\chi_{\mathrm{up}}(x_1,x_1'),\qquad \underline\chi_{\mathrm{lo}} = \inf_{x_1\ne x_1'}\chi_{\mathrm{lo}}(x_1,x_1')$$

such that for all large $n$,

$$\epsilon_n(\alpha, \underline\chi_{\mathrm{up}}) \le \epsilon_n^* \le \epsilon_n(\alpha, \underline\chi_{\mathrm{lo}}) \qquad\text{and}\qquad \delta_n = \frac{\alpha}{n}(1+o(1))$$

Thus, the privacy-locus $\epsilon_n^*$ is tightly sandwiched within an asymptotic band defined by $$(\underline\chi_{\mathrm{lo}}, \underline\chi_{\mathrm{up}})$$.

6. Practical Blanket-Divergence Accountant via FFT

Computing the blanket divergence for finite $n$ with controlled relative error $\eta>0$ and running time $\tilde O(n/\eta)$ proceeds by the following steps:

• Truncation: Restrict $l_\epsilon(Y)$ to $[s,s+w^{\mathrm{in}}]$ so that tail mass $$q=\Pr[l_\epsilon(Y)\notin[s,s+w^{\mathrm{in}}]] = O((n\gamma)^{-1}\eta)$$.
• Discretization: Discretize the truncated $Z^{\mathrm{tr}}$ on a mesh of width $h$, controlling mean-square error

$$\Delta = |\mathbb E[Z^{\mathrm{tr}}]-\mathbb E[Z^{\mathrm{di}}]| = O(\frac{\eta}{n\gamma})$$

and tail probabilities via Bernstein bounds.

• FFT Convolution: Zero-pad and FFT to compute the convolution of the PMF of $Z^{\mathrm{di}}$ with itself $(n-1)$ times at cost $O(N\log N)$, $N=w^{\mathrm{out}}/h$.
• Aggregate Probability: Recover the relevant tail via

$$\Pr[l_{\epsilon}(Y_1 )+ \sum_{i=2}^M Z_i> 0] = \Pr[\sum_{i=2}^M Z_i > -l_\epsilon(Y_1)]$$

and combine with the relevant measures $\mathcal R_{x_1}, \mathcal R_{x_1'}$.

Four error sources—truncation, discretization, aliasing, and CLT coupling—are tuned such that each contributes at most $(\eta/4) D$, yielding certified relative error $O(\eta)$. In moderate deviation regime, typical parameters satisfy $w^{\mathrm{in}} = \Theta(n^\alpha)$, $h = \Theta(c/\sqrt{n\log n})$, $w^{\mathrm{out}} = \Theta(\sqrt{n\log n})$, $N=O(n/\eta (\log n)^2)$, so total complexity is $\tilde O(n/\eta)$. The mid-point between one-sided bounds yields the final estimate.

Table: Blanket Divergence—Mechanism-Specific Parameters

Mechanism Type	Blanket Mass $\gamma$	Shuffle Index $\chi_{\mathrm{lo}}$
$k$-RR ($k\ge3$)	$kq,\;q=1/(e^{\epsilon_0}+k-1)$	$\sqrt{q/2(p-q)^2}$
Generalized Gaussian	$\int\inf_{x} \dots dy$	Numerical $\chi$ via variance calculation

The blanket divergence, in conjunction with the shuffle index and FFT-based numerical accounting, establishes a rigorous, tight framework for privacy analysis in the shuffle model beyond pure-LDP assumptions (Takagi et al., 27 Jan 2026).