A constrained risk inequality for general losses

John C. Duchi; Feng Ruan

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arXiv · 1804.08116

A constrained risk inequality for general losses

John C. Duchi·Feng Ruan

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Abstract

We provide a general constrained risk inequality that applies to arbitrary non-decreasing losses, extending a result of Brown and Low [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.

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John C. Duchi, Feng Ruan. 2018-04-22. A constrained risk inequality for general losses. https://arxiv.org/abs/1804.08116

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