On the Stability of Spherical Hellinger-Kantorovich Flows and Their Implications for Differential Privacy

We consider the privacy amplification properties of a sampling scheme in which a user's data isused in $k$ steps chosen randomly and uniformly from a sequence (or set) of $t$ steps. This sampling scheme has been recently applied in the context of differentially private optimization (Chua et al., 2024a; Choquette-Choo et al., 2025) and communication-efficient high-dimensional private aggregation (Asi et al., 2026), where it was shown to have utility advantages over the standard Poisson sampling. Theoretical analyses of this sampling scheme (Feldman & Shenfeld, 2025; Dong et al., 2025) lead to bounds that are close to those of Poisson sampling, yet still have two significant shortcomings. First, in many practical settings, the resulting privacy parameters are not tight due to the approximation steps in the analysis. Second, the computed parameters are either the hockey stick or Renyi divergence, both of which introduce overheads when used in privacy loss accounting. In this work, we demonstrate that the privacy loss distribution (PLD) of random allocation applied to any differentially private algorithm can be computed efficiently. When applied to the Gaussian mechanism, our results demonstrate that the privacy-utility trade-off for random allocation is at least as good as that of Poisson subsampling. In particular, random allocation is better suited for training via DP-SGD. To support these computations, our work develops new tools for general privacy loss accounting based on a notion of PLD realization. This notion allows us to extend accurate privacy loss accounting to subsampling which previously required manual noise-mechanism-specific analysis.

We derive a tight analysis of the trade-off function for Differentially Private Stochastic Gradient Descent (DP-SGD) with subsampling based on random shuffling within the $f$-DP framework. Our analysis covers the regime $σ\geq \sqrt{3/\ln M}$, where $σ$ is the noise multiplier and $M$ is the number of rounds within a single epoch. Unlike $f$-DP analyses for Poisson subsampling, which yield non-closed implicit formulas that can be machine computed but are non-transparent, random shuffling admits a tight analysis yielding transparent and interpretable closed-form bounds. Our concrete bounds, derived via the Berry-Esseen theorem, are tight up to constant factors within the proof framework. We demonstrate worked parameter settings for a single epoch ($E=1$) with a corresponding trade-off function $\geq 1-a-δ$, that is, only $δ$ below the ideal random guessing diagonal $1-a$: For $δ= 1/100$ and $σ= 1$, roughly $M \approx 1.14\times 10^6$ rounds and $N \approx 1.14\times 10^7$ training samples suffice to achieve meaningful differential privacy. This is in contrast to recent negative results for the regime $σ\leq 1/\sqrt{2 \ln M}$. Our concrete bounds can be composed over multiple epochs leading to $δ$ having a linear in $E$ dependency, which restricts $E=O(\sqrt{M})$. To go beyond Berry--Esseen, we introduce a new proof technique based on a generalization of the law of large numbers that yields an asymptotic random guessing diagonal-limit result: if $E=c_M^2M$ with $c_M\to 0$, then the $E$-fold composed trade-off function satisfies $f^{\otimes E}(a)\to 1-a$ uniformly in $a\in[0,1]$ with $δ$ having only an $O(\sqrt{E})$ dependency. We compare this asymptotic regime with the corresponding Poisson subsampling asymptotic, and highlight the characterization of explicit convergence rates as an open question.

In this paper, we study the problem of sampling from a distribution under the constraint of differential privacy (DP). Prior works measure the utility of DP sampling with density ratio-based measures such as KL divergence. However, such formulations suffer from two key limitations: 1) they fail to capture the geometric structure of the support, and 2) they are not applicable when the supports of the distributions differ. To deal with these issues, we develop a novel framework for DP sampling with Wasserstein distance as the utility measure. In this formulation, we propose Wasserstein Projection Mechanism (WPM), a minimax optimal mechanism based on Wasserstein projection. Furthermore, we develop efficient algorithms for computing the proposed mechanisms approximately and provide convergence guarantees.