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Online Distribution Learning with Local Privacy Constraints

Jin Sima · Changlong Wu · Olgica Milenkovic · Wojciech Szpankowski

MR1 & MR2 - Number 76
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Thu 2 May 8 a.m. PDT — 8:30 a.m. PDT

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.

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