We study the problem of sampling from a distribution under local differential privacy (LDP). Given a private distribution $P \in \mathcal{P}$, the goal is to generate a single sample from a distribution that remains close to $P$ in $f$-divergence while satisfying the constraints of LDP. This task captures the fundamental challenge of producing realistic-looking data under strong privacy guarantees. While prior work by Park et al. (NeurIPS'24) focuses on global minimax-optimality across a class of distributions, we take a local perspective. Specifically, we examine the minimax error in a neighborhood around a fixed distribution $P_0$, and characterize its exact value, which depends on both $P_0$ and the privacy level. Our main result shows that the local minimax error is determined by the global minimax error when the distribution class $\mathcal{P}$ is restricted to a neighborhood around $P_0$. To establish this, we (1) extend previous work from pure LDP to the more general functional LDP framework, and (2) prove that the globally optimal functional LDP sampler yields the optimal local sampler when constrained to distributions near $P_0$. Building on this, we also derive a simple closed-form expression for the locally minimax-optimal samplers which does not depend on the choice of $f$-divergence. We further argue that this local framework naturally models private sampling with public data, where the public data distribution is represented by $P_0$. In this setting, we empirically compare our locally optimal sampler to existing global methods, and demonstrate that it consistently outperforms global minimax samplers.

As the volume of data invested in statistical learning increases and concerns regarding privacy grow, the privacy leakage issue has drawn significant attention. Differential privacy has emerged as a widely accepted concept capable of mitigating privacy concerns, and numerous differentially private (DP) versions of machine learning algorithms have been developed. However, existing works on DP kernel learning algorithms have exhibited practical limitations, including scalability, restricted choice of kernels, or dependence on test data availability. We propose DP scalable kernel empirical risk minimization (ERM) algorithms and a DP kernel mean embedding (KME) release algorithm suitable for general kernels. Our approaches address the shortcomings of previous algorithms by employing Nyström methods, classical techniques in non-private scalable kernel learning. These methods provide data-dependent low-rank approximations of the kernel matrix for general kernels in a DP manner. We present excess empirical risk bounds and computational complexities for the scalable kernel DP ERM, KME algorithms, contrasting them with established methodologies. Furthermore, we develop a private data-generating algorithm capable of learning diverse kernel models. We conduct experiments to demonstrate the performance of our algorithms, comparing them with existing methods to highlight their superiority.

As concerns about data privacy continue to grow, differential privacy (DP) has emerged as a fundamental concept that aims to guarantee privacy by ensuring individuals' indistinguishability in data analysis. Local differential privacy (LDP) is a rigorous type of DP that requires individual data to be privatized before being sent to the collector, thus removing the need for a trusted third party to collect data. Among the numerous (L)DP-based approaches, functional DP has gained considerable attention in the DP community because it connects DP to statistical decision-making by formulating it as a hypothesis-testing problem and also exhibits Gaussian-related properties. However, the utility of privatized data is generally lower than that of non-private data, prompting research into optimal mechanisms that maximize the statistical utility for given privacy constraints. In this study, we investigate how functional LDP preserves the statistical utility by analyzing minimax risks of univariate mean estimation as well as nonparametric density estimation. We leverage the contraction property of functional LDP mechanisms and classical information-theoretical bounds to derive private minimax lower bounds. Our theoretical study reveals that it is possible to establish an interpretable, continuous balance between the statistical utility and privacy level, which has not been achieved under the $\epsilon$-LDP framework. Furthermore, we suggest minimax optimal mechanisms based on Gaussian LDP (a type of functional LDP) that achieve the minimax upper bounds and show via a numerical study that they are superior to the counterparts derived under $\epsilon$-LDP. The theoretical and empirical findings of this work suggest that Gaussian LDP should be considered a reliable standard for LDP.