Privately Learning Smooth Distributions on the Hypercube by Projections
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912030423777280 |
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| author | Lalanne, Clément Gadat, Sébastien |
| author_facet | Lalanne, Clément Gadat, Sébastien |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10083 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Privately Learning Smooth Distributions on the Hypercube by Projections Lalanne, Clément Gadat, Sébastien Statistics Theory 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. |
| title | Privately Learning Smooth Distributions on the Hypercube by Projections |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2409.10083 |