Online Distribution Learning with Local Privacy Constraints

Jin Sima, Changlong Wu, Olgica Milenkovic, Wojciech Szpankowski
Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR 238:460-468, 2024.

Abstract

We study the problem of online conditional distribution estimation with \emph{unbounded} label sets under local differential privacy. The problem may be succinctly stated as follows. Let $\mathcal{F}$ be a distribution-valued function class with an unbounded label set. Our aim is to estimate an \emph{unknown} function $f\in \mathcal{F}$ in an online fashion. More precisely, at time $t$, given a sample ${\mathbf{x}}_t$, we generate an estimate of $f({\mathbf{x}}_t)$ using only a \emph{privatized} version of the true \emph{labels} sampled from $f({\mathbf{x}}_t)$. The objective is to minimize the cumulative KL-risk of a finite horizon $T$. We show that under $(\epsilon,0)$-local differential privacy for the labels, the KL-risk equals $\tilde{\Theta}(\frac{1}{\epsilon}\sqrt{KT}),$ up to poly-logarithmic factors, where $K=|\mathcal{F}|$. This result significantly differs from the $\tilde{\Theta}(\sqrt{T\log K})$ bound derived in Wu et al., (2023a) for \emph{bounded} label sets. As a side-result, our approach recovers a nearly tight upper bound for the hypothesis selection problem of Gopi et al., (2020), which has only been established for the \emph{batch} setting.

Cite this Paper

BibTeX
@InProceedings{pmlr-v238-sima24a, title = { Online Distribution Learning with Local Privacy Constraints }, author = {Sima, Jin and Wu, Changlong and Milenkovic, Olgica and Szpankowski, Wojciech}, booktitle = {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics}, pages = {460--468}, year = {2024}, editor = {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen}, volume = {238}, series = {Proceedings of Machine Learning Research}, month = {02--04 May}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v238/sima24a/sima24a.pdf}, url = {https://proceedings.mlr.press/v238/sima24a.html}, abstract = { We study the problem of online conditional distribution estimation with \emph{unbounded} label sets under local differential privacy. The problem may be succinctly stated as follows. Let $\mathcal{F}$ be a distribution-valued function class with an unbounded label set. Our aim is to estimate an \emph{unknown} function $f\in \mathcal{F}$ in an online fashion. More precisely, at time $t$, given a sample ${\mathbf{x}}_t$, we generate an estimate of $f({\mathbf{x}}_t)$ using only a \emph{privatized} version of the true \emph{labels} sampled from $f({\mathbf{x}}_t)$. The objective is to minimize the cumulative KL-risk of a finite horizon $T$. We show that under $(\epsilon,0)$-local differential privacy for the labels, the KL-risk equals $\tilde{\Theta}(\frac{1}{\epsilon}\sqrt{KT}),$ up to poly-logarithmic factors, where $K=|\mathcal{F}|$. This result significantly differs from the $\tilde{\Theta}(\sqrt{T\log K})$ bound derived in Wu et al., (2023a) for \emph{bounded} label sets. As a side-result, our approach recovers a nearly tight upper bound for the hypothesis selection problem of Gopi et al., (2020), which has only been established for the \emph{batch} setting. } }
Endnote
%0 Conference Paper %T Online Distribution Learning with Local Privacy Constraints %A Jin Sima %A Changlong Wu %A Olgica Milenkovic %A Wojciech Szpankowski %B Proceedings of The 27th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2024 %E Sanjoy Dasgupta %E Stephan Mandt %E Yingzhen Li %F pmlr-v238-sima24a %I PMLR %P 460--468 %U https://proceedings.mlr.press/v238/sima24a.html %V 238 %X We study the problem of online conditional distribution estimation with \emph{unbounded} label sets under local differential privacy. The problem may be succinctly stated as follows. Let $\mathcal{F}$ be a distribution-valued function class with an unbounded label set. Our aim is to estimate an \emph{unknown} function $f\in \mathcal{F}$ in an online fashion. More precisely, at time $t$, given a sample ${\mathbf{x}}_t$, we generate an estimate of $f({\mathbf{x}}_t)$ using only a \emph{privatized} version of the true \emph{labels} sampled from $f({\mathbf{x}}_t)$. The objective is to minimize the cumulative KL-risk of a finite horizon $T$. We show that under $(\epsilon,0)$-local differential privacy for the labels, the KL-risk equals $\tilde{\Theta}(\frac{1}{\epsilon}\sqrt{KT}),$ up to poly-logarithmic factors, where $K=|\mathcal{F}|$. This result significantly differs from the $\tilde{\Theta}(\sqrt{T\log K})$ bound derived in Wu et al., (2023a) for \emph{bounded} label sets. As a side-result, our approach recovers a nearly tight upper bound for the hypothesis selection problem of Gopi et al., (2020), which has only been established for the \emph{batch} setting.
APA
Sima, J., Wu, C., Milenkovic, O. & Szpankowski, W.. (2024). Online Distribution Learning with Local Privacy Constraints . Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 238:460-468 Available from https://proceedings.mlr.press/v238/sima24a.html.

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