A Constrained Risk Inequality for General Losses
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
Text
Duchi and Ruan. "A Constrained Risk Inequality for General Losses." Artificial Intelligence and Statistics, 2021.Markdown
[Duchi and Ruan. "A Constrained Risk Inequality for General Losses." Artificial Intelligence and Statistics, 2021.](https://mlanthology.org/aistats/2021/duchi2021aistats-constrained/)BibTeX
@inproceedings{duchi2021aistats-constrained,
title = {{A Constrained Risk Inequality for General Losses}},
author = {Duchi, John and Ruan, Feng},
booktitle = {Artificial Intelligence and Statistics},
year = {2021},
pages = {802-810},
volume = {130},
url = {https://mlanthology.org/aistats/2021/duchi2021aistats-constrained/}
}