Log-Concave Duality in Estimation and Control

TL;DR

This paper extends the classical estimation-control duality to cases involving log-concave noise distributions, enabling new approaches for estimation with nonsmooth densities and establishing dual control problems.

Contribution

It generalizes the estimation-control duality to log-concave noise models and provides conditions for strong duality, including relaxed conditions for piecewise linear-quadratic cases.

Findings

Established duality for log-concave noise in estimation and control

Provided conditions for strong duality to hold

Demonstrated applications in nonsmooth density estimation

Abstract

In this paper we generalize the estimation-control duality that exists in the linear-quadratic-Gaussian setting. We extend this duality to maximum a posteriori estimation of the system's state, where the measurement and dynamical system noise are independent log-concave random variables. More generally, we show that a problem which induces a convex penalty on noise terms will have a dual control problem. We provide conditions for strong duality to hold, and then prove relaxed conditions for the piecewise linear-quadratic case. The results have applications in estimation problems with nonsmooth densities, such as log-concave maximum likelihood densities. We conclude with an example reconstructing optimal estimates from solutions to the dual control problem, which has implications for sharing solution methods between the two types of problems.