No-Free-Lunch Theorems in the continuum

TL;DR

This paper explores the applicability of No-Free-Lunch theorems in continuum domains, providing a simpler approach that clarifies the influence of domain structure and size on algorithm performance equivalence.

Contribution

It introduces a new, simpler method for analyzing No-Free-Lunch theorems in continuum domains, connecting discrete and continuous cases with fewer assumptions.

Findings

No-Free-Lunch theorems generally do not hold in continuum domains.

The new approach clarifies the role of domain cardinality and structure.

Results bridge the understanding between discrete and continuum optimization.

Abstract

No-Free-Lunch Theorems state, roughly speaking, that the performance of all search algorithms is the same when averaged over all possible objective functions. This fact was precisely formulated for the first time in a now famous paper by Wolpert and Macready, and then subsequently refined and extended by several authors, always in the context of a set of functions with discrete domain and codomain. Recently, Auger and Teytaud have shown that for continuum domains there is typically no No-Free-Lunch theorems. In this paper we provide another approach, which is simpler, requires less assumptions, relates the discrete and continuum cases, and that we believe that clarifies the role of the cardinality and structure of the domain.