# A Van Trees Lower Bound for Fully Interactive Differentially Private Federated Learning

arXiv: https://arxiv.org/abs/2605.19813 · HTML: https://mlsanity.com/paper/2605.19813/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 46 references that are in the map.
- First posted on arXiv 19 May 2026.

## Builds on (most cited first, 30 of 31)

- [Concentrated Differential Privacy: Simplifications, Extensions, and Lower Bounds](https://mlsanity.com/paper/1605.02065/) (2016)
- [Concentrated Differential Privacy](https://mlsanity.com/paper/1603.01887/) (2016)
- [Differentially Private Empirical Risk Minimization: Efficient Algorithms and Tight Error Bounds](https://mlsanity.com/paper/1405.7085/) (2014)
- [Minimax Optimal Procedures for Locally Private Estimation](https://mlsanity.com/paper/1604.02390/) (2016)
- [Fingerprinting Codes and the Price of Approximate Differential Privacy](https://mlsanity.com/paper/1311.3158/) (2013)
- [Differentially Private Assouad, Fano, and Le Cam](https://mlsanity.com/paper/2004.06830/) (2020)
- [New Lower Bounds for Private Estimation and a Generalized Fingerprinting Lemma](https://mlsanity.com/paper/2205.08532/) (2022)
- [Private Federated Learning Without a Trusted Server: Optimal Algorithms for Convex Losses](https://mlsanity.com/paper/2106.09779/) (2021)
- [Local Privacy, Data Processing Inequalities, and Statistical Minimax Rates](https://mlsanity.com/paper/1302.3203/) (2013)
- [On the Privacy-Robustness-Utility Trilemma in Distributed Learning](https://mlsanity.com/paper/2302.04787/) (2023)
- [Fisher information under local differential privacy](https://mlsanity.com/paper/2005.10783/) (2020)
- [Unified lower bounds for interactive high-dimensional estimation under information constraints](https://mlsanity.com/paper/2010.06562/) (2020)
- [Score Attack: A Lower Bound Technique for Optimal Differentially Private Learning](https://mlsanity.com/paper/2303.07152/) (2023)
- [Local Privacy and Minimax Bounds: Sharp Rates for Probability Estimation](https://mlsanity.com/paper/1305.6000/) (2013)
- [Discrete Distribution Estimation under User-level Local Differential Privacy](https://mlsanity.com/paper/2211.03757/) (2022)
- [Local differential privacy: Elbow effect in optimal density estimation and adaptation over Besov ellipsoids](https://mlsanity.com/paper/1903.01927/) (2019)
- [Simple Binary Hypothesis Testing under Local Differential Privacy and Communication Constraints](https://mlsanity.com/paper/2301.03566/) (2023)
- [Optimal Federated Learning for Nonparametric Regression with Heterogeneous Distributed Differential Privacy Constraints](https://mlsanity.com/paper/2406.06755/) (2024)
- [Federated Transfer Learning with Differential Privacy](https://mlsanity.com/paper/2403.11343/) (2024)
- [Smooth Lower Bounds for Differentially Private Algorithms via Padding-and-Permuting Fingerprinting Codes](https://mlsanity.com/paper/2307.07604/) (2023)
- [Rate Optimality and Phase Transition for User-Level Local Differential Privacy](https://mlsanity.com/paper/2405.11923/) (2024)
- [Federated Nonparametric Hypothesis Testing with Differential Privacy Constraints: Optimal Rates and Adaptive Tests](https://mlsanity.com/paper/2406.06749/) (2024)
- [Minimax And Adaptive Transfer Learning for Nonparametric Classification under Distributed Differential Privacy Constraints](https://mlsanity.com/paper/2406.20088/) (2024)
- [Optimal estimation in private distributed functional data analysis](https://mlsanity.com/paper/2412.06582/) (2024)
- [Improved Analysis of Sparse Linear Regression in Local Differential Privacy Model](https://mlsanity.com/paper/2310.07367/) (2023)
- [Federated PCA and Estimation for Spiked Covariance Matrices: Optimal Rates and Efficient Algorithm](https://mlsanity.com/paper/2411.15660/) (2024)
- [Optimal Cox regression under federated differential privacy: coefficients and cumulative hazards](https://mlsanity.com/paper/2508.19640/) (2025)
- [Differentially Private Federated Learning: Servers Trustworthiness, Estimation, and Statistical Inference](https://mlsanity.com/paper/2404.16287/) (2024)
- [Optimal Federated Learning for Functional Mean Estimation under Heterogeneous Privacy Constraints](https://mlsanity.com/paper/2412.18992/) (2024)
- [Nonparametric spectral density estimation using interactive mechanisms under local differential privacy](https://mlsanity.com/paper/2504.00919/) (2025)

## Abstract (arXiv)

Federated differentially private protocols can communicate over many adaptive rounds and reuse each client's local samples. Existing lower bound arguments for federated DP are often restricted to noninteractive protocols or fresh batch decompositions, so the fundamental information-theoretic limit of estimation under fully interactive protocols remains unknown. We establish a federated van Trees inequality for parameter estimation under squared ℓ2 loss from any complete public transcript satisfying a clientwise zCDP constraint at the sample level. A scalar trace form covers homogeneous experiments, while a matrix form preserves directional Fisher geometry in heterogeneous experiments where different clients are informative in different subspaces. Together with existing upper bounds for the corresponding problems, these results identify the minimax rates for various statistical problems including mean estimation, linear regression, nonparametric regression, and functional mean estimation over the full class of interactive public-transcript protocols. For these problems, arbitrary public interaction and repeated sample reuse do not improve the rate over simpler restricted protocols. The key technical ingredient in our paper is a contraction inequality for the Fisher information in the transcript: each client's contribution is bounded both by the Fisher information in its local experiment and by its total privacy budget.