Online Distribution Learning with Local Private Constraints
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911768897388544 |
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| author | Sima, Jin Wu, Changlong Milenkovic, Olgica Szpankowski, Wojciech |
| author_facet | Sima, Jin Wu, Changlong Milenkovic, Olgica Szpankowski, Wojciech |
| contents | We study the problem of online conditional distribution estimation with \emph{unbounded} label sets under local differential privacy. Let $\mathcal{F}$ be a distribution-valued function class with unbounded label set. We aim at estimating an \emph{unknown} function $f\in \mathcal{F}$ in an online fashion so that at time $t$ when the context $\boldsymbol{x}_t$ is provided we can generate an estimate of $f(\boldsymbol{x}_t)$ under KL-divergence knowing only a privatized version of the true labels sampling from $f(\boldsymbol{x}_t)$. The ultimate objective is to minimize the cumulative KL-risk of a finite horizon $T$. We show that under $(ε,0)$-local differential privacy of the privatized labels, the KL-risk grows as $\tildeΘ(\frac{1}ε\sqrt{KT})$ upto poly-logarithmic factors where $K=|\mathcal{F}|$. This is in stark contrast to the $\tildeΘ(\sqrt{T\log K})$ bound demonstrated by Wu et al. (2023a) for bounded label sets. As a byproduct, our results recover a nearly tight upper bound for the hypothesis selection problem of gopi et al. (2020) established only for the batch setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_00315 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Online Distribution Learning with Local Private Constraints Sima, Jin Wu, Changlong Milenkovic, Olgica Szpankowski, Wojciech Machine Learning Cryptography and Security Data Structures and Algorithms Information Theory We study the problem of online conditional distribution estimation with \emph{unbounded} label sets under local differential privacy. Let $\mathcal{F}$ be a distribution-valued function class with unbounded label set. We aim at estimating an \emph{unknown} function $f\in \mathcal{F}$ in an online fashion so that at time $t$ when the context $\boldsymbol{x}_t$ is provided we can generate an estimate of $f(\boldsymbol{x}_t)$ under KL-divergence knowing only a privatized version of the true labels sampling from $f(\boldsymbol{x}_t)$. The ultimate objective is to minimize the cumulative KL-risk of a finite horizon $T$. We show that under $(ε,0)$-local differential privacy of the privatized labels, the KL-risk grows as $\tildeΘ(\frac{1}ε\sqrt{KT})$ upto poly-logarithmic factors where $K=|\mathcal{F}|$. This is in stark contrast to the $\tildeΘ(\sqrt{T\log K})$ bound demonstrated by Wu et al. (2023a) for bounded label sets. As a byproduct, our results recover a nearly tight upper bound for the hypothesis selection problem of gopi et al. (2020) established only for the batch setting. |
| title | Online Distribution Learning with Local Private Constraints |
| topic | Machine Learning Cryptography and Security Data Structures and Algorithms Information Theory |
| url | https://arxiv.org/abs/2402.00315 |