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A constrained risk inequality for general losses

John Duchi · Feng Ruan

Keywords: [ Algorithms ] [ Nonlinear Dimensionality Reduction and Manifold Learning ] [ Optimization ] [ Non-Convex Optimization ] [ Learning Theory and Statistics ] [ Decision Theory ]

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.

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