Optimal Polynomial-Time Estimators: A Bayesian Notion of Approximation Algorithm

TL;DR

This paper introduces optimal polynomial-time estimators, a Bayesian-inspired framework for approximation in decision problems, formalizing how limited-resource reasoners estimate solutions with probabilistic methods.

Contribution

It formalizes the concept of Bayesian-inspired estimators within average-case complexity, proving existence, completeness, and parallels with classical probability theory.

Findings

Existence theorems for optimal estimators

Completeness results linking estimators to complexity classes

Establishes parallels between estimators and classical probability theory

Abstract

We introduce a new concept of approximation applicable to decision problems and functions, inspired by Bayesian probability. From the perspective of a Bayesian reasoner with limited computational resources, the answer to a problem that cannot be solved exactly is uncertain and therefore should be described by a random variable. It thus should make sense to talk about the expected value of this random variable, an idea we formalize in the language of average-case complexity theory by introducing the concept of "optimal polynomial-time estimators." We prove some existence theorems and completeness results, and show that optimal polynomial-time estimators exhibit many parallels with "classical" probability theory.