# Unified lower bounds for interactive high-dimensional estimation under information constraints

arXiv: https://arxiv.org/abs/2010.06562 · HTML: https://mlsanity.com/paper/2010.06562/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Oct 2020.

## Built on by (most cited first, 22 of 22)

- [Discrete Distribution Estimation under User-level Local Differential Privacy](https://mlsanity.com/paper/2211.03757/) (2022)
- [Contraction of Locally Differentially Private Mechanisms](https://mlsanity.com/paper/2210.13386/) (2022)
- [The Role of Interactivity in Structured Estimation](https://mlsanity.com/paper/2203.06870/) (2022)
- [On the Statistical Complexity of Estimation and Testing under Privacy Constraints](https://mlsanity.com/paper/2210.02215/) (2022)
- [Order-Optimal Sequential 1-Bit Mean Estimation in General Tail Regimes](https://mlsanity.com/paper/2604.07796/) (2026)
- [Robust Estimation of Discrete Distributions under Local Differential Privacy](https://mlsanity.com/paper/2202.06825/) (2022)
- [Sequential 1-bit Mean Estimation with Near-Optimal Sample Complexity](https://mlsanity.com/paper/2509.21940/) (2025)
- [Improved Analysis of Sparse Linear Regression in Local Differential Privacy Model](https://mlsanity.com/paper/2310.07367/) (2023)
- [Fundamental limits of over-the-air optimization: Are analog schemes optimal?](https://mlsanity.com/paper/2109.05222/) (2021)
- [Private Statistical Estimation of Many Quantiles](https://mlsanity.com/paper/2302.06943/) (2023)
- [Sample-Optimal Locally Private Hypothesis Selection and the Provable Benefits of Interactivity](https://mlsanity.com/paper/2312.05645/) (2023)
- [Locally Private Sampling with Public Data](https://mlsanity.com/paper/2411.08791/) (2024)
- [Non-Stochastic CDF Estimation Using Threshold Queries](https://mlsanity.com/paper/2301.05682/) (2023)
- [About the Cost of Central Privacy in Density Estimation](https://mlsanity.com/paper/2306.14535/) (2023)
- [Differentially Private Federated Learning: Servers Trustworthiness, Estimation, and Statistical Inference](https://mlsanity.com/paper/2404.16287/) (2024)
- [Open Problem: Is Interaction Necessary for Order-Optimal 1-bit Mean Estimation?](https://mlsanity.com/paper/2607.02896/) (2026)
- [Bayesian Frequency Estimation Under Local Differential Privacy With an Adaptive Randomized Response Mechanism](https://mlsanity.com/paper/2405.07020/) (2024)
- [Better Locally Private Sparse Estimation Given Multiple Samples Per User](https://mlsanity.com/paper/2408.04313/) (2024)
- [Optimal Federated Learning for Functional Mean Estimation under Heterogeneous Privacy Constraints](https://mlsanity.com/paper/2412.18992/) (2024)
- [Uniformity Testing in the Shuffle Model: Simpler, Better, Faster](https://mlsanity.com/paper/2108.08987/) (2021)
- [A Van Trees Lower Bound for Fully Interactive Differentially Private Federated Learning](https://mlsanity.com/paper/2605.19813/) (2026)
- [Information-Theoretic Bounds for Sparse Covariance Estimation in the Vertical-Split Distributed Model](https://mlsanity.com/paper/2606.07124/) (2026)

## Abstract (arXiv)

We consider distributed parameter estimation using interactive protocols subject to local information constraints such as bandwidth limitations, local differential privacy, and restricted measurements. We provide a unified framework enabling us to derive a variety of (tight) minimax lower bounds for different parametric families of distributions, both continuous and discrete, under any ℓp loss. Our lower bound framework is versatile and yields "plug-and-play" bounds that are widely applicable to a large range of estimation problems, and, for the prototypical case of the Gaussian family, circumvents limitations of previous techniques. In particular, our approach recovers bounds obtained using data processing inequalities and Cram'er--Rao bounds, two other alternative approaches for proving lower bounds in our setting of interest. Further, for the families considered, we complement our lower bounds with matching upper bounds.