# Optimal Accounting of Differential Privacy via Characteristic Function

arXiv: https://arxiv.org/abs/2106.08567 · HTML: https://mlsanity.com/paper/2106.08567/

## Key facts

- Cited by 86 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 54 references that are in the map.
- First posted on arXiv 16 Jun 2021.

## Builds on (most cited first, 16 of 16)

- [Deep Learning with Differential Privacy](https://mlsanity.com/paper/1607.00133/) (2016)
- [Renyi Differential Privacy](https://mlsanity.com/paper/1702.07476/) (2017)
- [Concentrated Differential Privacy: Simplifications, Extensions, and Lower Bounds](https://mlsanity.com/paper/1605.02065/) (2016)
- [The Composition Theorem for Differential Privacy](https://mlsanity.com/paper/1311.0776/) (2013)
- [RAPPOR: Randomized Aggregatable Privacy-Preserving Ordinal Response](https://mlsanity.com/paper/1407.6981/) (2014)
- [Privacy Amplification by Subsampling: Tight Analyses via Couplings and Divergences](https://mlsanity.com/paper/1807.01647/) (2018)
- [Subsampled R'enyi Differential Privacy and Analytical Moments Accountant](https://mlsanity.com/paper/1808.00087/) (2018)
- [Improving the Gaussian Mechanism for Differential Privacy: Analytical Calibration and Optimal Denoising](https://mlsanity.com/paper/1805.06530/) (2018)
- [Numerical Composition of Differential Privacy](https://mlsanity.com/paper/2106.02848/) (2021)
- [The Discrete Gaussian for Differential Privacy](https://mlsanity.com/paper/2004.00010/) (2020)
- [Differentially Private Empirical Risk Minimization: Efficient Algorithms and Tight Error Bounds](https://mlsanity.com/paper/1405.7085/) (2014)
- [Hypothesis Testing Interpretations and Renyi Differential Privacy](https://mlsanity.com/paper/1905.09982/) (2019)
- [Privacy for Free: Posterior Sampling and Stochastic Gradient Monte Carlo](https://mlsanity.com/paper/1502.07645/) (2015)
- [Tight Differential Privacy for Discrete-Valued Mechanisms and for the Subsampled Gaussian Mechanism Using FFT](https://mlsanity.com/paper/2006.07134/) (2020)
- [Three Variants of Differential Privacy: Lossless Conversion and Applications](https://mlsanity.com/paper/2008.06529/) (2020)
- [Computing Differential Privacy Guarantees for Heterogeneous Compositions Using FFT](https://mlsanity.com/paper/2102.12412/) (2021)

## Built on by (most cited first, 30 of 86)

- [Privacy Auditing with One (1) Training Run](https://mlsanity.com/paper/2305.08846/) (2023)
- [Automatic Clipping: Differentially Private Deep Learning Made Easier and Stronger](https://mlsanity.com/paper/2206.07136/) (2022)
- [Differentially Private Optimization on Large Model at Small Cost](https://mlsanity.com/paper/2210.00038/) (2022)
- [Connect the Dots: Tighter Discrete Approximations of Privacy Loss Distributions](https://mlsanity.com/paper/2207.04380/) (2022)
- [Bounding Training Data Reconstruction in DP-SGD](https://mlsanity.com/paper/2302.07225/) (2023)
- [Composition of Differential Privacy & Privacy Amplification by Subsampling](https://mlsanity.com/paper/2210.00597/) (2022)
- [Private Fine-tuning of Large Language Models with Zeroth-order Optimization](https://mlsanity.com/paper/2401.04343/) (2024)
- [Dynamic Differential-Privacy Preserving SGD](https://mlsanity.com/paper/2111.00173/) (2021)
- [Auditing f-Differential Privacy in One Run](https://mlsanity.com/paper/2410.22235/) (2024)
- [Fine-Tuning Large Language Models with User-Level Differential Privacy](https://mlsanity.com/paper/2407.07737/) (2024)
- [FLUTE: A Scalable, Extensible Framework for High-Performance Federated Learning Simulations](https://mlsanity.com/paper/2203.13789/) (2022)
- [Fully Adaptive Composition in Differential Privacy](https://mlsanity.com/paper/2203.05481/) (2022)
- [How Private are DP-SGD Implementations?](https://mlsanity.com/paper/2403.17673/) (2024)
- [Individual Privacy Accounting with Gaussian Differential Privacy](https://mlsanity.com/paper/2209.15596/) (2022)
- [Practical Differentially Private Hyperparameter Tuning with Subsampling](https://mlsanity.com/paper/2301.11989/) (2023)
- [Privacy Amplification for Matrix Mechanisms](https://mlsanity.com/paper/2310.15526/) (2023)
- [Scalable DP-SGD: Shuffling vs. Poisson Subsampling](https://mlsanity.com/paper/2411.04205/) (2024)
- [Faster Privacy Accounting via Evolving Discretization](https://mlsanity.com/paper/2207.04381/) (2022)
- [Bounding data reconstruction attacks with the hypothesis testing interpretation of differential privacy](https://mlsanity.com/paper/2307.03928/) (2023)
- [Privacy amplification by random allocation](https://mlsanity.com/paper/2502.08202/) (2025)
- [Edgeworth Accountant: An Analytical Approach to Differential Privacy Composition](https://mlsanity.com/paper/2206.04236/) (2022)
- [Balls-and-Bins Sampling for DP-SGD](https://mlsanity.com/paper/2412.16802/) (2024)
- [Unified Enhancement of Privacy Bounds for Mixture Mechanisms via f-Differential Privacy](https://mlsanity.com/paper/2310.19973/) (2023)
- [Avoiding Pitfalls for Privacy Accounting of Subsampled Mechanisms under Composition](https://mlsanity.com/paper/2405.20769/) (2024)
- [Improving the Privacy and Practicality of Objective Perturbation for Differentially Private Linear Learners](https://mlsanity.com/paper/2401.00583/) (2023)
- [A Randomized Approach for Tight Privacy Accounting](https://mlsanity.com/paper/2304.07927/) (2023)
- [Subsampling is not Magic: Why Large Batch Sizes Work for Differentially Private Stochastic Optimisation](https://mlsanity.com/paper/2402.03990/) (2024)
- [Differentially Private Bootstrap: New Privacy Analysis and Inference Strategies](https://mlsanity.com/paper/2210.06140/) (2022)
- [Privacy Loss of Noisy Stochastic Gradient Descent Might Converge Even for Non-Convex Losses](https://mlsanity.com/paper/2305.09903/) (2023)
- [InvisibleInk: High-Utility and Low-Cost Text Generation with Differential Privacy](https://mlsanity.com/paper/2507.02974/) (2025)

## Abstract (arXiv)

Characterizing the privacy degradation over compositions, i.e., privacy accounting, is a fundamental topic in differential privacy (DP) with many applications to differentially private machine learning and federated learning. We propose a unification of recent advances (Renyi DP, privacy profiles, f-DP and the PLD formalism) via the characteristic function (φ-function) of a certain dominating privacy loss random variable. We show that our approach allows natural adaptive composition like Renyi DP, provides exactly tight privacy accounting like PLD, and can be (often losslessly) converted to privacy profile and f-DP, thus providing (ε,δ)-DP guarantees and interpretable tradeoff functions. Algorithmically, we propose an analytical Fourier accountant that represents the complex logarithm of φ-functions symbolically and uses Gaussian quadrature for numerical computation. On several popular DP mechanisms and their subsampled counterparts, we demonstrate the flexibility and tightness of our approach in theory and experiments.