Correcting Convexity Bias in Function and Functional Estimate

TL;DR

This paper introduces a general framework with methods to correct convexity bias in noisy function estimates, improving accuracy across various applications without requiring problem-specific knowledge.

Contribution

It proposes a versatile bias correction framework for convex function estimation that reduces error without needing detailed problem information.

Findings

Methods significantly reduce estimation error compared to baseline.

Applicable to diverse problems including optimization and distribution functionals.

Numerical experiments demonstrate improved estimate quality.

Abstract

A general framework with a series of different methods is proposed to improve the estimate of convex function (or functional) values when only noisy observations of the true input are available. Technically, our methods catch the bias introduced by the convexity and remove this bias from a baseline estimate. Theoretical analysis are conducted to show that the proposed methods can strictly reduce the expected estimate error under mild conditions. When applied, the methods require no specific knowledge about the problem except the convexity and the evaluation of the function. Therefore, they can serve as off-the-shelf tools to obtain good estimate for a wide range of problems, including optimization problems with random objective functions or constraints, and functionals of probability distributions such as the entropy and the Wasserstein distance. Numerical experiments on a wide variety of problems show that our methods can significantly improve the quality of the estimate compared with the baseline method.