Evolvability need not imply learnability

TL;DR

This paper demonstrates that certain Boolean functions are evolvable in a PAC sense with fixed hypothesis size, but this does not imply they are PAC-learnable, highlighting differences between evolvability and learnability.

Contribution

It provides a counter-example showing evolvability does not necessarily imply PAC-learnability for monotone Boolean functions.

Findings

Boolean functions as monotone DNF are PAC-evolvable with fixed hypothesis size

Counter-example disproves that evolvability implies PAC-learnability

Discussion of conditions where evolvability and learnability coincide or differ

Abstract

We show that Boolean functions expressible as monotone disjunctive normal forms are PAC-evolvable under a uniform distribution on the Boolean cube if the hypothesis size is allowed to remain fixed. We further show that this result is insufficient to prove the PAC-learnability of monotone Boolean functions, thereby demonstrating a counter-example to a recent claim to the contrary. We further discuss scenarios wherein evolvability and learnability will coincide as well as scenarios under which they differ. The implications of the latter case on the prospects of learning in complex hypothesis spaces is briefly examined.