Private Evaluation of High-Dimensional Functions
Abstract
We study differentially private black-box evaluation of arbitrary high-dimensional functions. Given a dataset S and query access to a function f---whose sensitivity may be unknown or unbounded---the goal is to privately output an approximation to f(S). The black-box evaluator can only query the value of f over a down neighborhood of S (i.e., those subsets of S that are obtained by removing a bounded number of elements from S). We design generic black-box evaluators for both pure and approximate differential privacy through a reduction to high-dimensional private quasi-concave optimization (QCO). Our black-box evaluator enjoys a geometric notion of utility. Consider the vectors obtained by applying f on sets in a down neighborhood of S. With high probability, the evaluator's output is guaranteed to be in the convex hull of these vectors. Our work generalizes the result of Linder et al. [LRSS25] for real-valued function evaluation to high dimensions. It also recovers the private center-point guarantees of Beimel et al. [BMNS19], with an explicit additive approximation to maximum Tukey depth. We also apply our evaluators to user-level quasi-concave and convex optimization, showing how they can improve upon the sample-and-aggregate baseline of Nissim et al. [NRS07] by replacing the multiplicative data overhead with an additive overhead. Finally, we show that our formulation of high dimensional black-box evaluation and high dimensional private QCO are equivalent up to a factor of the dimension: the hardest private evaluation instances already arise from the cases that f is a (non-private) quasi-concave optimizer.
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BibTeX
@misc{afzali2026private,
title = {{Private Evaluation of High-Dimensional Functions}},
author = {Mohammad Afzali and Hassan Ashtiani and Adam Smith},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24477},
eprint = {2610.24477},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24477},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Private Evaluation of High-Dimensional Functions
AU - Mohammad Afzali
AU - Hassan Ashtiani
AU - Adam Smith
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24477
UR - https://arxiv.org/abs/2610.24477
AB - We study differentially private black-box evaluation of arbitrary high-dimensional functions. Given a dataset S and query access to a function f---whose sensitivity may be unknown or unbounded---the goal is to privately output an approximation to f(S). The black-box evaluator can only query the value of f over a down neighborhood of S (i.e., those subsets of S that are obtained by removing a bounded number of elements from S). We design generic black-box evaluators for both pure and approximate differential privacy through a reduction to high-dimensional private quasi-concave optimization (QCO). Our black-box evaluator enjoys a geometric notion of utility. Consider the vectors obtained by applying f on sets in a down neighborhood of S. With high probability, the evaluator's output is guaranteed to be in the convex hull of these vectors. Our work generalizes the result of Linder et al. [LRSS25] for real-valued function evaluation to high dimensions. It also recovers the private center-point guarantees of Beimel et al. [BMNS19], with an explicit additive approximation to maximum Tukey depth. We also apply our evaluators to user-level quasi-concave and convex optimization, showing how they can improve upon the sample-and-aggregate baseline of Nissim et al. [NRS07] by replacing the multiplicative data overhead with an additive overhead. Finally, we show that our formulation of high dimensional black-box evaluation and high dimensional private QCO are equivalent up to a factor of the dimension: the hardest private evaluation instances already arise from the cases that f is a (non-private) quasi-concave optimizer.
ER -