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About the Cost of Central Privacy in Density Estimation

Clément Lalanne, Aurélien Garivier, Rémi Gribonval

TL;DR

This work studies non-parametric density estimation for densities in Lipschitz and Sobolev spaces, and under central privacy, and extends it to other norms and notions of privacy, finding that the estimation is optimal.

Abstract

We study non-parametric density estimation for densities in Lipschitz and Sobolev spaces, and under central privacy. In particular, we investigate regimes where the privacy budget is not supposed to be constant. We consider the classical definition of central differential privacy, but also the more recent notion of central concentrated differential privacy. We recover the result of Barber and Duchi (2014) stating that histogram estimators are optimal against Lipschitz distributions for the L2 risk, and under regular differential privacy, and we extend it to other norms and notions of privacy. Then, we investigate higher degrees of smoothness, drawing two conclusions: First, and contrary to what happens with constant privacy budget (Wasserman and Zhou, 2010), there are regimes where imposing privacy degrades the regular minimax risk of estimation on Sobolev densities. Second, so-called projection estimators are near-optimal against the same classes of densities in this new setup with pure differential privacy, but contrary to the constant privacy budget case, it comes at the cost of relaxation. With zero concentrated differential privacy, there is no need for relaxation, and we prove that the estimation is optimal.

About the Cost of Central Privacy in Density Estimation

TL;DR

This work studies non-parametric density estimation for densities in Lipschitz and Sobolev spaces, and under central privacy, and extends it to other norms and notions of privacy, finding that the estimation is optimal.

Abstract

We study non-parametric density estimation for densities in Lipschitz and Sobolev spaces, and under central privacy. In particular, we investigate regimes where the privacy budget is not supposed to be constant. We consider the classical definition of central differential privacy, but also the more recent notion of central concentrated differential privacy. We recover the result of Barber and Duchi (2014) stating that histogram estimators are optimal against Lipschitz distributions for the L2 risk, and under regular differential privacy, and we extend it to other norms and notions of privacy. Then, we investigate higher degrees of smoothness, drawing two conclusions: First, and contrary to what happens with constant privacy budget (Wasserman and Zhou, 2010), there are regimes where imposing privacy degrades the regular minimax risk of estimation on Sobolev densities. Second, so-called projection estimators are near-optimal against the same classes of densities in this new setup with pure differential privacy, but contrary to the constant privacy budget case, it comes at the cost of relaxation. With zero concentrated differential privacy, there is no need for relaxation, and we prove that the estimation is optimal.
Paper Structure (45 sections, 10 theorems, 88 equations, 1 table)

This paper contains 45 sections, 10 theorems, 88 equations, 1 table.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions
  4. Central Differential Privacy
  5. A quick word on local privacy.
  6. Histogram Estimators and Lipschitz Densities
  7. General utility of histogram estimators
  8. Privacy and bin size tuning
  9. Lower-bounds and minimax optimality
  10. General framework.
  11. Pointwise lower-bound
  12. On the optimality and on the cost of privacy.
  13. Integrated lower-bound
  14. Projection Estimators and Periodic Sobolev Densities
  15. General utility of projection estimators
  16. ...and 30 more sections

Key Result

Lemma 1

[lemma]lemma:histogramutility There exists $C_L > 0$, a positive constant that only depends on $L$, such that

Theorems & Definitions (15)

  • Lemma 1: General utility of \ref{['eq:hist']}
  • Theorem 1: Privacy and utility of \ref{['eq:hist']} - DP case
  • Theorem 2: Pointwise lower-bound
  • proof : Proof idea
  • Corollary 1: Infinite norm lower-bound
  • Theorem 3: Integrated lower-bound
  • proof : Proof idea
  • Lemma 2: General utility of \ref{['def:projestimator']}
  • Theorem 4: Privacy and utility of \ref{['def:projestimator']}
  • Theorem 5: Integrated lower-bound
  • ...and 5 more