Privately Learning Smooth Distributions on the Hypercube by Projections

Clément Lalanne; Sébastien Gadat

Home/Archive/arXiv

arXiv · 2409.10083

Privately Learning Smooth Distributions on the Hypercube by Projections

Clément Lalanne·Sébastien Gadat

View original source ↗Read source document ↗Export metadata ↓

Abstract

Fueled by the ever-increasing need for statistics that guarantee the privacy of their training sets, this article studies the centrally-private estimation of Sobolev-smooth densities of probability over the hypercube in dimension d. The contributions of this article are two-fold : Firstly, it generalizes the one dimensional results of (Lalanne et al., 2023) to non-integer levels of smoothness and to a high-dimensional setting, which is important for two reasons : it is more suited for modern learning tasks, and it allows understanding the relations between privacy, dimensionality and smoothness, which is a central question with differential privacy. Secondly, this article presents a private strategy of estimation that is data-driven (usually referred to as adaptive in Statistics) in order to privately choose an estimator that achieves a good bias-variance trade-off among a finite family of private projection estimators without prior knowledge of the ground-truth smoothness $β$. This is achieved by adapting the Lepskii method for private selection, by adding a new penalization term that makes the estimation privacy-aware.

Explore related subjects

math.ST stat.TH

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Clément Lalanne, Sébastien Gadat. 2024-09-16. Privately Learning Smooth Distributions on the Hypercube by Projections. https://arxiv.org/abs/2409.10083

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Transitional Conditional Independence

Statistical models contain variables that are not random: parameters, treatments, environments, design points. Ordinary conditional independence cannot express relations involving such variables. To apply it one must first put a distribution on them, and that changes the meaning of the statement. This paper introduces transitional conditional independence. It relates three variables on a Markov kernel $K(W|T)$ with non-stochastic input $T$, and is defined by a single factorization: \[ X\perp\!\!\perp_{K(W|T)} Y |Z \quad :\iff \quad \exists\, Q(X|Z):\; K(X,Y,Z|T) = Q(X|Z)\otimes K(Y,Z|T).\] The relation asserts a Markov kernel $Q(X|Z)$ that is the same for every input $t$. It therefore yields a factorization rather than an almost-sure identity between conditional expectations, and it needs no distribution on the input space. The relation is asymmetric. We show that the asymmetry is essential: symmetrizing it destroys the statements it was built to make. We prove left and right versions of all separoid rules except Symmetry. Ten of them hold on arbitrary measurable spaces, the remaining ones under one condition on the spaces involved, and we give criteria for when Symmetry itself holds. We axiomatize the resulting structure and show that it arises from any symmetric separoid by a shift. We give several applications. Ancillarity, sufficiency and adequacy become factorizations that hold pointwise in the parameter, without a prior and without null sets; the theorems of Fisher--Neyman and of Basu take this form. The invariance hypothesis of invariant prediction, $Y \perp\!\!\perp E | X_S$, receives its intended meaning: one kernel predicts $Y$ from $X_S$ in every environment $E$. And Bayesian networks with non-stochastic input nodes satisfy a directed global Markov property whose graphical id-separation criterion returns a factorization of Markov kernels, on arbitrary input spaces.

math.ST↗

arXiv2026

Mean Residual Life Ageing Intensity Function

Ageing intensity is usually formulated through the failure rate, whereas its mean residual life ($MRL$)-based counterpart has not been systematically developed. This paper introduces the mean residual life ageing intensity ($MRLAI$) function as an $MRL$-based analogue of the failure-rate-based ageing intensity ($FRAI$) function. The proposed function compares the current $MRL$ with its average over $[0,t]$, providing a relative measure of residual-life ageing behaviour. Basic properties of the $MRLAI$ function are established, including a characterisation of the exponential distribution. The decreasing and increasing $MRLAI$ classes are introduced and related to the usual $MRL$-based ageing classes. Examples are used to clarify the interplay among $FRAI$, $MRLAI$, and $MRL$ monotonicity. Closure and non-closure properties of the proposed classes are examined under mixtures, convolutions, and system formation. An $MRLAI$-based stochastic order is also defined, characterised, and compared with classical stochastic orders, and its preservation properties under selected system operations and transformations are investigated.

math.ST↗

arXiv2026

Self-Normalizing Denominators in Rational Covariance Estimators

Many estimators are ratios of coprime polynomials in a sample covariance matrix, and their accuracy depends on the relative fluctuation of the sample denominator. Under Gaussian sampling in fixed dimension, we call a nonconstant polynomial denominator self-normalizing if the first-order variance of its relative error does not depend on the population covariance. We prove that these denominators are exactly the flag powers, nonzero constant multiples of products of positive integer powers of nested generalized variances. Equivalently, the denominator's sample-to-population ratio has a covariance-independent finite-sample law, which we determine explicitly. Sufficiency is classical; the new converse shows that a first-order variance condition forces an exact sampling law. We show that relative stability, meaning bounded first-order relative variance, characterizes uniform tightness of scaled relative errors over positive-definite covariances. It permits replacing the sample denominator by its population value in the limit theory of the ratio. Self-normalization is its rigid core. We locate these classes in applications, where regression on predecessors in a fixed order yields only constant or self-normalizing denominators, instrumental-variable formulas yield relatively unstable ones, and nonparametric identifiability does not guarantee relative stability.

math.ST↗