Optimal Algorithms for Nonlinear Estimation with Convex Models

TL;DR

This paper extends optimal estimation techniques from linear functionals to certain nonlinear functionals within convex models, using convex optimization and advanced functional analysis tools.

Contribution

It introduces a method to optimally estimate nonlinear functionals, such as maxima of linear functionals, within convex symmetric sets, expanding classical linear estimation theory.

Findings

Optimal estimation of nonlinear functionals achieved.

Convex optimization provides practical estimation algorithms.

Refined Hahn-Banach theorem supports theoretical results.

Abstract

A linear functional of an object from a convex symmetric set can be optimally estimated, in a worst-case sense, by a linear functional of observations made on the object. This well-known fact is extended here to a nonlinear setting: other simple functionals of the object can be optimally estimated by functionals of the observations that share a similar simple structure. This is established for the maximum of several linear functionals and even for the $\ell$th largest among them. Proving the latter requires an unusual refinement of the analytical Hahn--Banach theorem. The existence results are accompanied by practical recipes relying on convex optimization to construct the desired functionals, thereby justifying the term of estimation algorithms.