Title: Locally Optimal Private Sampling: Beyond the Global Minimax
Venue: NeurIPS 2025
Accepted: yes
Keywords: private sampling

local minimax

local differential privacy

functional LDP
Abstract: 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.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies how to generate private samples from a user’s data under local differential privacy (LDP).  In particular, the authors show that the locally minimax-optimal private sampler can be derived from the global solution restricted to a local neighborhood. They provide explicit formulas for these samplers and prove they are optimal for a broad class of privacy definitions. Experiments on both discrete and continuous data show that the proposed local samplers outperform existing global methods in terms of accuracy across various privacy levels.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nThe paper is well-organized and logically structured, with a clear progression from global to local minimax analysis. This work makes a contribution to the field of private sampling by introducing a local minimax-optimal framework that extends beyond traditional worst-case analysis.\n\nWeaknesses:\n\n1. The condition $\\frac{c_2 - c_1}{1 - c_1} \\in \\mathbb{N}$ in Assumption 3.2 is somewhat restrictive and lacks clear justification, potentially limiting the generality of the results. What is the practical impact of requiring $\\\\frac{c_2 - c_1}{1 - c_1} \\in \\mathbb{N}$? Can you demonstrate that this condition approximately holds for common distributions or suggest a relaxation?\n2. The intuition behind the superiority of the non-linear sampler, currently deferred to the appendix, would be more impactful if summarized earlier in the main manuscript to guide the reader’s understanding.\n3. The practical utility of the proposed method remains unclear due to the lack of experiments on real-world datasets. Including a discussion on potential applications, such as private estimation or generative modeling, would help contextualize the broader impact. \n4. While the neighborhood-based formulation builds on prior work, the connection to existing literature in local differential privacy estimation is not sufficiently emphasized. \n5. In high-dimensional settings, does the clipping technique outlined in Theorem 5.1 become computationally prohibitive? It would be helpful to discuss the associated complexity and whether efficient approximations or relaxations are feasible.\n6. The relationship between the proposed local minimax framework and instance-optimal differential privacy (e.g., Feldman et al. [44]) is not explicitly addressed. Clarifying this connection could provide further insight, particularly regarding implications for private SGD or generative modeling.\n\nQUESTIONS:\nPlease see the Weaknesses.\n\nLIMITATIONS:\nPlease see the Weaknesses.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-30"}

comments: SUMMARY:
In this paper, the authors generalize the optimal minimax LDP sampling procedure of Park to the functional-LDP setting.
Then, they consider distributions that are neighbors of a fixed distribution $P_0$ and present 2 algorithms to perform sampling in the pure LDP setting. Lastly, authors compare their 2 algorithms with synthetic experiments.

STRENGTHS AND WEAKNESSES:
Strengths
- The paper is well written, the hypotheses are appropriately discussed, and the results statements are clear.
- The generalization of the work of Park to the functional LDP setting is an interesting and new contribution
- The existence of a minimax optimal sampler that is point-wise better than the one derived from the generalization of the work of Park is also an interesting contribution.  

Weaknesses
- The class of distributions for which the sampler is optimal is restricted (the density verifies: c1 h(x) <= p(x) <= c2 h(x)). Although some restrictions are necessary to exclude trivial samplers, it is unclear to me why this particular condition arises or whether other choices could have been made.

QUESTIONS:
Could you discuss the choice of distribution class ?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-01

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates the problem of distribution sampling under local differential privacy (LDP). The authors begin by examining the global setting, where they extend existing results to both approximate LDP and functional LDP. They then shift focus to the local setting, in which the distribution class is restricted to a neighborhood around a fixed distribution $P_0$. Building on their global sampling approach, they develop a sampler that achieves the optimal minimax rate in the local setting.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nThe paper addresses an important and timely problem in distribution sampling under local differential privacy, and it proposes a solution that achieves the optimal minimax rate. The extension from the global to the local setting is meaningful and adds significant value to the literature.\n\nWeaknesses and Questions:\n\n1. The presentation could be improved, particularly regarding the purpose and organization of Section 3. Although the paper primarily focuses on local sampling, Section 3 is entirely devoted to global sampling. While I understand parts of section 3 is necessary for later sections like extension to functional LDP case and introducing $Q^*_g$, it would be clearer to restructure this material into a dedicated subsection.\n2. The proposed sampler in Theorem 4.1 is not computation efficient for general $\\widetilde{\\mathcal{P}}$. In particular, the projection step to compute $\\hat{P}$ can be computation demanding. By the way, what's the meaning of \"minimizes $\\inf_{P'\\in N(P_0)} D_f(P||P')$\"? For any fixed $P$ and $P_0$, this infimum is a constant value.\n3. The relationship between global and local sampling is not fully clarified. Intuitively. one would assume that since local sampler is restricted to a smaller distribution set, it will achieve a better minimax rate. However, this improvement is not quantified in the paper. If general case is infeasible to quantify, is it possible to at least give concrete examples to demonstrate the rate gap between global and local sampling.\n\nThere are some other minor comments:\n1. what is the definition of $g^*$ in equation 4?\n2. The definition of $\\widetilde{P}$ is unclear in section 4, are we still considering class defined in equation (2) or a general distribution class is considered?\n\nQUESTIONS:\nThe questions are included in the \"Strengths And Weaknesses\" section.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the problem of generating a sample from a\ndistribution under the constraints of local differential privacy.\nThis work extends previous work in two directions: 1) it studies\nsampling algorithms satisfying functional local differential privacy\nwith optimal global minimax risk 2) it studies local minimax risks of\nprivate samplers in the presence of public data. The proposed\nalgorithms are validate experimentally.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\n- The paper presents two methods to identify private samplers that are\n  locally minimax optimal when a public distribution is available,\n\n- the paper characterizes samplers that are globally minimax optimal\n  and that satisfy functional LDP.\n\nWeaknesses\n\n- The focus of the paper on the local model makes the sampling problem\n  less about algorithms and more about finding the right parameters,\n\n- the paper doesn't give much intuition about the derived samplers.\n\nSeveral recent works have studied sampling from distributions under\nthe constraints of differential privacy in different models. This\npaper considers this problem in the local model of differential\nprivacy, where every user holds a distribution and aims at releasing a\nsample from it. More specifically, the paper focuses on minmax notions\nof utility of the samplers with respect to f-divergences. In\nparticular, the utility is defined at the user level rather than at\nthe protocol level as in works that have studied statistical tasks in\nthe local model. While certainly meaningful, this focus makes the\npaper more about figuring out the right setup and parameters of \nthe samplers rather than coming up with novel algorithm.  \nThe paper does this well focusing on problems that were not studied \nin previous works. The global minimax results extend the set of results available to\nfunctional LDP and to different discrete and continuous\nsamplers. Similarly to what I said above, functional DP was proposed\nin the context of the central model.  While it can certainly also be\nused for the local model, its properties and semantics are different\nand make some problem more interesting and some problem less\ninteresting. The results on local minimax extend previous work which\nstudied samplers with public data in the local model by providing\noptimal local minimax samplers. It is worth to notice that the paper\npresents two frameworks to do this. The experiments confirm the\ntheoretical results. Overall, I fund this paper contributing good\nresults to the theory of differentially private local samplers,\nalthough I didn't find them particularly exciting in terms of new\nideas required.\n\nQUESTIONS:\nCan you please provide a comparison between how you use public data an previous work [22] used them?\n\nIn the limitation you mentions the computational complexity of your samplers, can you give more details on it?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-23"}