Concept class

In computational learning theory in mathematics, a concept over a domain X is a total Boolean function over X. A concept class is a class of concepts. Concept classes are a subject of computational learning theory.

Concept class terminology frequently appears in model theory associated with probably approximately correct (PAC) learning.^[1] In this setting, if one takes a set Y as a set of (classifier output) labels, and X is a set of examples, the map ${\displaystyle c:X\to Y}$, i.e. from examples to classifier labels (where ${\displaystyle Y=\{0,1\}}$ and where c is a subset of X), c is then said to be a concept. A concept class ${\displaystyle C}$ is then a collection of such concepts.

Given a class of concepts C, a subclass D is reachable if there exists a sample s such that D contains exactly those concepts in C that are extensions to s.^[2] Not every subclass is reachable.^[2][why?]

Background

A sample ${\displaystyle s}$ is a partial function from ${\displaystyle X}$^{[clarification needed]} to ${\displaystyle \{0,1\}}$.^[2] Identifying a concept with its characteristic function mapping ${\displaystyle X}$ to ${\displaystyle \{0,1\}}$, it is a special case of a sample.^[2]

Two samples are consistent if they agree on the intersection of their domains.^[2] A sample ${\displaystyle s'}$ extends another sample ${\displaystyle s}$ if the two are consistent and the domain of ${\displaystyle s}$ is contained in the domain of ${\displaystyle s'}$.^[2]

Examples

Suppose that ${\displaystyle C=S^{+}(X)}$. Then:

• the subclass ${\displaystyle \{\{x\}\}}$ is reachable with the sample ${\displaystyle s=\{(x,1)\}}$;^[2][why?]
• the subclass ${\displaystyle S^{+}(Y)}$ for ${\displaystyle Y\subseteq X}$ are reachable with a sample that maps the elements of ${\displaystyle X-Y}$ to zero;^[2][why?]
• the subclass ${\displaystyle S(X)}$, which consists of the singleton sets, is not reachable.^[2][why?]

Applications

Let ${\displaystyle C}$ be some concept class. For any concept ${\displaystyle c\in C}$, we call this concept ${\displaystyle 1/d}$-good for a positive integer ${\displaystyle d}$ if, for all ${\displaystyle x\in X}$, at least ${\displaystyle 1/d}$ of the concepts in ${\displaystyle C}$ agree with ${\displaystyle c}$ on the classification of ${\displaystyle x}$.^[2] The fingerprint dimension ${\displaystyle FD(C)}$ of the entire concept class ${\displaystyle C}$ is the least positive integer ${\displaystyle d}$ such that every reachable subclass ${\displaystyle C'\subseteq C}$ contains a concept that is ${\displaystyle 1/d}$-good for it.^[2] This quantity can be used to bound the minimum number of equivalence queries^{[clarification needed]} needed to learn a class of concepts according to the following inequality:${\textstyle FD(C)-1\leq \#EQ(C)\leq \lceil FD(C)\ln(|C|)\rceil }$.^[2]

References