Bounding the Fat Shattering Dimension of a Composition Function Class Built Using a Continuous Logic Connective

We begin this report by describing the Probably Approximately Correct (PAC) model for learning a concept class, consisting of subsets of a domain, and a function class, consisting of functions from the domain to the unit interval. Two combinatorial parameters, the Vapnik-Chervonenkis (VC) dimension and its generalization, the Fat Shattering dimension of scale e, are explained and a few examples of their calculations are given with proofs. We then explain Sauer’s Lemma, which involves the VC dimension and is used to prove the equivalence of a concept class being distribution-free PAC learnable and it having finite VC dimension. As the main new result of our research, we explore the construction of a new function class, obtained by forming compositions with a continuous logic connective, a uniformly continuous function from the unit hypercube to the unit interval, from a collection of function classes. Vidyasagar had proved that such a composition function class has finite Fat Shattering dimension of all scales if the classes in the original collection do; however, no estimates of the dimension were known. Using results by Mendelson-Vershynin and Talagrand, we bound the Fat Shattering dimension of scale e of this new function class in terms of the Fat Shattering dimensions of the collection’s classes. We conclude this report by providing a few open questions and future research topics involving the PAC learning model.

💡 Research Summary

The paper provides a thorough exposition of the Probably Approximately Correct (PAC) learning framework, beginning with the classic definitions of concept classes (subsets of a domain) and function classes (real‑valued functions mapping a domain into the unit interval). It reviews the two combinatorial complexity measures that govern learnability: the Vapnik‑Chervonenkis (VC) dimension for binary‑valued concepts and its real‑valued analogue, the Fat Shattering dimension at scale ε. After recalling Sauer’s Lemma and its role in proving that a concept class is distribution‑free PAC learnable if and only if its VC dimension is finite, the authors turn to the less‑explored territory of Fat Shattering dimension for function classes.

The novel contribution of the work lies in analyzing how the Fat Shattering dimension behaves under a composition operation defined by a continuous logic connective. A continuous logic connective is a uniformly continuous function u :