Learning Multivariate Log-concave Distributions

Ilias Diakonikolas; Daniel M. Kane; Alistair Stewart

Learning Multivariate Log-concave Distributions

Ilias Diakonikolas, Daniel M. Kane, Alistair Stewart

Proceedings of the 2017 Conference on Learning Theory, PMLR 65:711-727, 2017.

Abstract

We study the problem of estimating multivariate log-concave probability density functions. We prove the first sample complexity upper bound for learning log-concave densities on

$\mathbb{R}^d$ , for all

$d ≥1$ . Prior to our work, no upper bound on the sample complexity of this learning problem was known for the case of

$d>3$ . In more detail, we give an estimator that, for any

$d \ge 1$ and

$ε>0$ , draws

$\tilde{O}_d \left( (1/ε)^(d+5)/2 \right)$ samples from an unknown target log-concave density on

$R^d$ , and outputs a hypothesis that (with high probability) is

$ε$ -close to the target, in total variation distance. Our upper bound on the sample complexity comes close to the known lower bound of

$\Omega_d \left( (1/ε)^(d+1)/2 \right)$ for this problem.

Cite this Paper

BibTeX


@InProceedings{pmlr-v65-diakonikolas17a,
  title = 	 {Learning Multivariate Log-concave Distributions},
  author = 	 {Diakonikolas, Ilias and Kane, Daniel M. and Stewart, Alistair},
  booktitle = 	 {Proceedings of the 2017 Conference on Learning Theory},
  pages = 	 {711--727},
  year = 	 {2017},
  editor = 	 {Kale, Satyen and Shamir, Ohad},
  volume = 	 {65},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {07--10 Jul},
  publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v65/diakonikolas17a/diakonikolas17a.pdf},
  url = 	 {https://proceedings.mlr.press/v65/diakonikolas17a.html},
  abstract = 	 {We study the problem of estimating multivariate log-concave probability density functions. We prove the first sample complexity upper bound for learning log-concave densities on $\mathbb{R}^d$, for all $d ≥1$. Prior to our work, no upper bound on the sample complexity of this learning problem was known for the case of $d>3$. In more detail, we give an estimator that, for any $d \ge 1$ and $ε>0$, draws $\tilde{O}_d \left( (1/ε)^(d+5)/2 \right)$ samples from an unknown target log-concave density on $R^d$, and outputs a hypothesis that (with high probability) is $ε$-close to the target, in total variation distance. Our upper bound on the sample complexity comes close to the known lower bound of $\Omega_d \left( (1/ε)^(d+1)/2 \right)$ for this problem. }
}

Endnote

%0 Conference Paper
%T Learning Multivariate Log-concave Distributions
%A Ilias Diakonikolas
%A Daniel M. Kane
%A Alistair Stewart
%B Proceedings of the 2017 Conference on Learning Theory
%C Proceedings of Machine Learning Research
%D 2017
%E Satyen Kale
%E Ohad Shamir	
%F pmlr-v65-diakonikolas17a
%I PMLR
%P 711--727
%U https://proceedings.mlr.press/v65/diakonikolas17a.html
%V 65
%X We study the problem of estimating multivariate log-concave probability density functions. We prove the first sample complexity upper bound for learning log-concave densities on $\mathbb{R}^d$, for all $d ≥1$. Prior to our work, no upper bound on the sample complexity of this learning problem was known for the case of $d>3$. In more detail, we give an estimator that, for any $d \ge 1$ and $ε>0$, draws $\tilde{O}_d \left( (1/ε)^(d+5)/2 \right)$ samples from an unknown target log-concave density on $R^d$, and outputs a hypothesis that (with high probability) is $ε$-close to the target, in total variation distance. Our upper bound on the sample complexity comes close to the known lower bound of $\Omega_d \left( (1/ε)^(d+1)/2 \right)$ for this problem.

APA


Diakonikolas, I., Kane, D.M. & Stewart, A.. (2017). Learning Multivariate Log-concave Distributions. Proceedings of the 2017 Conference on Learning Theory, in Proceedings of Machine Learning Research 65:711-727 Available from https://proceedings.mlr.press/v65/diakonikolas17a.html.

Related Material

Download PDF