Lower bounds for the minimax risk using $f$-divergences and applications

TL;DR

This paper develops a unified framework using $f$-divergences to derive minimax risk lower bounds in estimation problems, simplifying proofs and extending known inequalities with applications to convex body reconstruction and covariance matrix estimation.

Contribution

It introduces a general approach leveraging $f$-divergences for minimax lower bounds, unifying and simplifying existing inequalities and providing new bounds with practical applications.

Findings

Unified $f$-divergence framework for minimax bounds

Simplified proofs using convexity properties

New lower bounds for convex body reconstruction and covariance estimation

Abstract

Lower bounds involving $f$-divergences between the underlying probability measures are proved for the minimax risk in estimation problems. Our proofs just use simple convexity facts. Special cases and straightforward corollaries of our bounds include well known inequalities for establishing minimax lower bounds such as Fano's inequality, Pinsker's inequality and inequalities based on global entropy conditions. Two applications are provided: a new minimax lower bound for the reconstruction of convex bodies from noisy support function measurements and a different proof of a recent minimax lower bound for the estimation of a covariance matrix.