Minimax density estimation in the adversarial framework under local differential privacy

Mélisande Albert (1, 2) , Juliette Chevallier (1, 2) , Béatrice Laurent (2, 1) , Ousmane Sacko (3, 4)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 INSA Toulouse - Institut National des Sciences Appliquées - Toulouse
3 UPN - Université Paris Nanterre
4 MODAL'X - Modélisation aléatoire de Paris X

We consider the problem of nonparametric density estimation under privacy constraints in an adversarial framework. To this end, we study minimax rates under local differential privacy over Sobolev spaces. We first obtain a lower bound which allows us to quantify the impact of privacy compared with the classical framework. Next, we introduce a new Coordinate block privacy mechanism that guarantees local differential privacy, which, coupled with a projection estimator, achieves the minimax optimal rates.

Mots clés: Nonparametric density estimation; local differential privacy; minimax optimality; adversarial loss; Sobolev balls

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Mélisande Albert, Juliette Chevallier, Béatrice Laurent, Ousmane Sacko. Minimax density estimation in the adversarial framework under local differential privacy. 2024. ⟨hal-04522328⟩