Adaptive Confidence Sets for Nonparametric Density Estimation under Local Differential Privacy
Abstract
In nonparametric density estimation without privacy constraints, it is known that a confidence set whose coverage holds uniformly over a `rough' class of regularity r≥0 can simultaneously attain, in L2-diameter, the minimax rate over a smaller, smoother class of regularity s>r if and only if s ≤ 2r. We study such adaptive honest confidence sets under non-interactive α-local differential privacy (LDP) constraints. Our main result is that, for bounded α, such `smooth-rate adaptation' is now possible if and only if s ≤ 4r/3. In this smaller range we construct an honest private confidence set whose diameter is of the private minimax order on each of the two classes. As in the non-private case, the threshold arises because an honest set cannot, at a smooth density, be narrower than the separation rate arising in a related testing problem. Under LDP we prove that this private testing rate is of order (nα2)−2r/(4r+3), which is at most the private estimation rate (nα2)−s/(2s+2) over the smooth class exactly when s ≤ 4r/3. For s > 4r/3, however, we show that smooth-rate adaptation becomes possible once one discards from the rough class the densities that lie outside the smooth class but within an L2-radius of the order of the private testing rate.
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BibTeX
@misc{khan2026adaptive,
title = {{Adaptive Confidence Sets for Nonparametric Density Estimation under Local Differential Privacy}},
author = {Aditya Khan and Thibault Randrianarisoa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15982},
eprint = {2610.15982},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15982},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Adaptive Confidence Sets for Nonparametric Density Estimation under Local Differential Privacy
AU - Aditya Khan
AU - Thibault Randrianarisoa
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15982
UR - https://arxiv.org/abs/2610.15982
AB - In nonparametric density estimation without privacy constraints, it is known that a confidence set whose coverage holds uniformly over a `rough' class of regularity r≥0 can simultaneously attain, in L^2-diameter, the minimax rate over a smaller, smoother class of regularity s>r if and only if s ≤ 2r. We study such adaptive honest confidence sets under non-interactive α-local differential privacy (LDP) constraints. Our main result is that, for bounded α, such `smooth-rate adaptation' is now possible if and only if s ≤ 4r/3. In this smaller range we construct an honest private confidence set whose diameter is of the private minimax order on each of the two classes. As in the non-private case, the threshold arises because an honest set cannot, at a smooth density, be narrower than the separation rate arising in a related testing problem. Under LDP we prove that this private testing rate is of order (nα^2)^−2r/(4r+3), which is at most the private estimation rate (nα^2)^−s/(2s+2) over the smooth class exactly when s ≤ 4r/3. For s > 4r/3, however, we show that smooth-rate adaptation becomes possible once one discards from the rough class the densities that lie outside the smooth class but within an L^2-radius of the order of the private testing rate.
ER -