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Optimal Algorithms for Mean Estimation under Local Differential Privacy
Machine Learning · Submitted 5 May 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that PrivUnit (Bhowmick et al. 2018) with optimized parameters achieves the optimal variance among a large family of locally private randomizers.
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The abstract, sentence by sentence
We study the problem of mean estimation of $\ell_2$-bounded vectors under the constraint of local differential privacy. While the literature has a variety of algorithms that achieve the asymptotically optimal rates for this problem, the performance of these algorithms in practice can vary significantly due to varying (and often large) hidden constants. In this work, we investigate the question of designing the protocol with the smallest variance. We show that PrivUnit (Bhowmick et al. 2018) with optimized parameters achieves the optimal variance among a large family of locally private randomizers. To prove this result, we establish some properties of local randomizers, and use symmetrization arguments that allow us to write the optimal randomizer as the optimizer of a certain linear program. These structural results, which should extend to other problems, then allow us to show that the optimal randomizer belongs to the PrivUnit family. We also develop a new variant of PrivUnit based on the Gaussian distribution which is more amenable to mathematical analysis and enjoys the same optimality guarantees. This allows us to establish several useful properties on the exact constants of the optimal error as well as to numerically estimate these constants.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study the problem of mean estimation of $\ell_2$-bounded vectors under the constraint of local differential privacy. | background 0.92 | 0.38 | 0.02 | asserted 0.72 |
| s1 | While the literature has a variety of algorithms that achieve the asymptotically optimal rates for this problem, the performance of these algorithms in practice can vary significantly due to varying (and often large) hidden constants. | gap 0.72 | 0.08 | 0.02 | asserted 0.96 |
| s2 | In this work, we investigate the question of designing the protocol with the smallest variance. | contribution 0.89 | 0.47 | 0.05 | unclear 0.57 |
| s3 | We show that PrivUnit (Bhowmick et al. 2018) with optimized parameters achieves the optimal variance among a large family of locally private randomizers. | result 0.98 | 0.85 | 0.88 | asserted 1.00 |
| s4 | To prove this result, we establish some properties of local randomizers, and use symmetrization arguments that allow us to write the optimal randomizer as the optimizer of a certain linear program. | method 0.98 | 0.95 | 0.04 | asserted 1.00 |
| s5 | These structural results, which should extend to other problems, then allow us to show that the optimal randomizer belongs to the PrivUnit family. | result 0.75 | 0.92 | 0.23 | asserted 0.99 |
| s6 | We also develop a new variant of PrivUnit based on the Gaussian distribution which is more amenable to mathematical analysis and enjoys the same optimality guarantees. | contribution 0.83 | 0.96 | 0.06 | asserted 1.00 |
| s7 | This allows us to establish several useful properties on the exact constants of the optimal error as well as to numerically estimate these constants. | result 0.68 | 0.94 | 0.05 | asserted 0.98 |
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Original abstract
We study the problem of mean estimation of $\ell_2$-bounded vectors under the constraint of local differential privacy. While the literature has a variety of algorithms that achieve the asymptotically optimal rates for this problem, the performance of these algorithms in practice can vary significantly due to varying (and often large) hidden constants. In this work, we investigate the question of designing the protocol with the smallest variance. We show that PrivUnit (Bhowmick et al. 2018) with optimized parameters achieves the optimal variance among a large family of locally private randomizers. To prove this result, we establish some properties of local randomizers, and use symmetrization arguments that allow us to write the optimal randomizer as the optimizer of a certain linear program. These structural results, which should extend to other problems, then allow us to show that the optimal randomizer belongs to the PrivUnit family. We also develop a new variant of PrivUnit based on the Gaussian distribution which is more amenable to mathematical analysis and enjoys the same optimality guarantees. This allows us to establish several useful properties on the exact constants of the optimal error as well as to numerically estimate these constants.
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How the literature uses it
4 arXiv papers cite this in 6 sentences
Papers citing it · 3 sentences without a confident label
Cites it as support 1 paper
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Hence, block design mechanisms with appropriate $k$ also achieve the maximum mutual information. * Second, in the proof of showing the optimality of PrivUnit$_2$ [ref] for mean estimation in [ref] , it was shown that given a scheme, its `symmetrized' version gives a lower or equal worst-case risk.
Exactly Optimal and Communication-Efficient Private Estimation via Block Designs · Other decomposable loss functions · 0.97
Cites it as background 2 papers
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Many tight results are known for mean estimation [ref] , convex ERM optimization [ref] with specific solution restrictions like on $l_1$ norm [ref] , stochastic optimization [ref] , and principal component analysis (PCA) [ref] .
PAC Privacy: Automatic Privacy Measurement and Control of Data Processing · Conclusions and Prospects · 0.95
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Finally, many works, such as [ref] , study mean estimation, with and without compression, under local DP [ref] .
Private Federated Learning with Autotuned Compression · Introduction · 0.98
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