A constrained risk inequality for general losses

John Duchi, Feng Ruan
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, PMLR 130:802-810, 2021.

Abstract

We provide a general constrained risk inequality that applies to arbitrary non-decreasing losses, extending a result of Brown and Low [\emph{Ann. Stat. 1996}]. Given two distributions $P_0$ and $P_1$, we find a lower bound for the risk of estimating a parameter $\theta(P_1)$ under $P_1$ given an upper bound on the risk of estimating the parameter $\theta(P_0)$ under $P_0$. The inequality is a useful pedagogical tool, as its proof relies only on the Cauchy-Schwartz inequality, it applies to general losses, and it transparently gives risk lower bounds on super-efficient and adaptive estimators.

Cite this Paper

BibTeX
@InProceedings{pmlr-v130-duchi21a, title = { A constrained risk inequality for general losses }, author = {Duchi, John and Ruan, Feng}, booktitle = {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics}, pages = {802--810}, year = {2021}, editor = {Banerjee, Arindam and Fukumizu, Kenji}, volume = {130}, series = {Proceedings of Machine Learning Research}, month = {13--15 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v130/duchi21a/duchi21a.pdf}, url = {https://proceedings.mlr.press/v130/duchi21a.html}, abstract = { We provide a general constrained risk inequality that applies to arbitrary non-decreasing losses, extending a result of Brown and Low [\emph{Ann. Stat. 1996}]. Given two distributions $P_0$ and $P_1$, we find a lower bound for the risk of estimating a parameter $\theta(P_1)$ under $P_1$ given an upper bound on the risk of estimating the parameter $\theta(P_0)$ under $P_0$. The inequality is a useful pedagogical tool, as its proof relies only on the Cauchy-Schwartz inequality, it applies to general losses, and it transparently gives risk lower bounds on super-efficient and adaptive estimators. } }
Endnote
%0 Conference Paper %T A constrained risk inequality for general losses %A John Duchi %A Feng Ruan %B Proceedings of The 24th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2021 %E Arindam Banerjee %E Kenji Fukumizu %F pmlr-v130-duchi21a %I PMLR %P 802--810 %U https://proceedings.mlr.press/v130/duchi21a.html %V 130 %X We provide a general constrained risk inequality that applies to arbitrary non-decreasing losses, extending a result of Brown and Low [\emph{Ann. Stat. 1996}]. Given two distributions $P_0$ and $P_1$, we find a lower bound for the risk of estimating a parameter $\theta(P_1)$ under $P_1$ given an upper bound on the risk of estimating the parameter $\theta(P_0)$ under $P_0$. The inequality is a useful pedagogical tool, as its proof relies only on the Cauchy-Schwartz inequality, it applies to general losses, and it transparently gives risk lower bounds on super-efficient and adaptive estimators.
APA
Duchi, J. & Ruan, F.. (2021). A constrained risk inequality for general losses . Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 130:802-810 Available from https://proceedings.mlr.press/v130/duchi21a.html.

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