Privacy amplification by random allocation

TL;DR

This paper provides the first theoretical analysis and numerical estimation methods for privacy amplification in a random k-out-of-t sampling scheme, improving privacy guarantees in differential privacy applications.

Contribution

It introduces the first theoretical bounds and efficient estimation algorithms for privacy amplification by random allocation, connecting it to known subsampling techniques.

Findings

Bounds the privacy guarantees of random k-out-of-t sampling using independent subsampling.

Provides numerical algorithms for estimating privacy guarantees in practical scenarios.

Demonstrates near-tight bounds for Gaussian noise addition in privacy amplification.

Abstract

We consider the privacy amplification properties of a sampling scheme in which a user's data is used 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 is also motivated by communication-efficient high-dimensional private aggregation [Asi et al., 2025]. Existing analyses of this scheme either rely on privacy amplification by shuffling which leads to overly conservative bounds or require Monte Carlo simulations that are computationally prohibitive in most practical scenarios. We give the first theoretical guarantees and numerical estimation algorithms for this sampling scheme. In particular, we demonstrate that the privacy guarantees of random k-out-of-t allocation can be upper bounded by the privacy guarantees of the well-studied independent (or Poisson) subsampling in which each step uses the user's data with probability $(1+o(1))k/t$. Further, we provide two additional analysis techniques that lead to numerical improvements in several parameter regimes. Altogether, our bounds give efficiently-computable and nearly tight numerical results for random allocation applied to Gaussian noise addition.