arXiv subjects

Mohammad Afzali

Publications and source records attributed to Mohammad Afzali.

5 recordsLinked to original sources

arXiv2026

Actually, All Learnable Distribution Classes Are Privately Learnable

We show that every class of distributions that is learnable up to arbitrarily small target error in total variation distance with finitely many samples is also learnable under ($\varepsilon, δ$)-differential privacy up to arbitrarily small target error with finitely many samples. This proves a conjecture of Ashtiani. Previous work by Bun, Kamath, Mouzakis, and Singhal [arXiv:2402.00267] gave a weak refutation of this conjecture, showing that this implication is not true when learnability is defined with respect to a fixed target error. In contrast, we show that, under the standard definition of learnability, the conjecture is true.

cs.LG↗

arXiv2026

Private Evaluation of High-Dimensional Functions

We study differentially private black-box evaluation of arbitrary high-dimensional functions. Given a dataset $S$ and query access to a function $f$---whose sensitivity may be unknown or unbounded---the goal is to privately output an approximation to $f(S)$. The black-box evaluator can only query the value of $f$ over a down neighborhood of $S$ (i.e., those subsets of $S$ that are obtained by removing a bounded number of elements from $S$). We design generic black-box evaluators for both pure and approximate differential privacy through a reduction to high-dimensional private quasi-concave optimization (QCO). Our black-box evaluator enjoys a geometric notion of utility. Consider the vectors obtained by applying $f$ on sets in a down neighborhood of $S$. With high probability, the evaluator's output is guaranteed to be in the convex hull of these vectors. Our work generalizes the result of Linder et al. [LRSS25] for real-valued function evaluation to high dimensions. It also recovers the private center-point guarantees of Beimel et al. [BMNS19], with an explicit additive approximation to maximum Tukey depth. We also apply our evaluators to user-level quasi-concave and convex optimization, showing how they can improve upon the sample-and-aggregate baseline of Nissim et al. [NRS07] by replacing the multiplicative data overhead with an additive overhead. Finally, we show that our formulation of high dimensional black-box evaluation and high dimensional private QCO are equivalent up to a factor of the dimension: the hardest private evaluation instances already arise from the cases that $f$ is a (non-private) quasi-concave optimizer.

stat.ML↗

arXiv2026

Efficient Agnostic Learning of Univariate Densities Under Local Differential Privacy

We propose a general framework for non-interactive locally differentially private (LDP) learning of structured univariate distributions that can be approximated by piece-wise polynomials. This gives the first sample-efficient LDP learners for several distribution classes, including mixtures of log-concave distributions, such as Gaussian mixture models (GMMs). These learners also provide agnostic guarantees and are computationally efficient. We also prove a lower bound for learning GMMs under LDP, improving on the standard pure-DP packing lower bounds of Bun et al. [BKSW19]. To obtain our general learning framework, we develop an efficient non-interactive LDP algorithm for synthetic data generation over large domains, which may be of independent interest. We then use it together with a deterministic adaptation of the non-private piecewise polynomial learner of Acharya et al. [ADLS17].

stat.ML↗

arXiv2024

Agnostic Private Density Estimation for GMMs via List Global Stability

We consider the problem of private density estimation for mixtures of unrestricted high dimensional Gaussians in the agnostic setting. We prove the first upper bound on the sample complexity of this problem. Previously, private learnability of high dimensional GMMs was only known in the realizable setting [Afzali et al., 2024]. To prove our result, we exploit the notion of $\textit{list global stability}$ [Ghazi et al., 2021b,a] that was originally introduced in the context of private supervised learning. We define an agnostic variant of this definition, showing that its existence is sufficient for agnostic private density estimation. We then construct an agnostic list globally stable learner for GMMs.

stat.ML↗

arXiv2024

Mixtures of Gaussians are Privately Learnable with a Polynomial Number of Samples

We study the problem of estimating mixtures of Gaussians under the constraint of differential privacy (DP). Our main result is that $\text{poly}(k,d,1/α,1/\varepsilon,\log(1/δ))$ samples are sufficient to estimate a mixture of $k$ Gaussians in $\mathbb{R}^d$ up to total variation distance $α$ while satisfying $(\varepsilon, δ)$-DP. This is the first finite sample complexity upper bound for the problem that does not make any structural assumptions on the GMMs. To solve the problem, we devise a new framework which may be useful for other tasks. On a high level, we show that if a class of distributions (such as Gaussians) is (1) list decodable and (2) admits a "locally small'' cover (Bun et al., 2021) with respect to total variation distance, then the class of its mixtures is privately learnable. The proof circumvents a known barrier indicating that, unlike Gaussians, GMMs do not admit a locally small cover (Aden-Ali et al., 2021b).

stat.ML↗