Computer Science – Learning
Scientific paper
2010-06-26
Lect. Notes in Artificial Intelligence 6331, Springer, 2010, pp. 134-147
Computer Science
Learning
14 pages, 1 figure, latex 2e with Springer macros
Scientific paper
In response to a 1997 problem of M. Vidyasagar, we state a necessary and sufficient condition for distribution-free PAC learnability of a concept class $\mathscr C$ under the family of all non-atomic (diffuse) measures on the domain $\Omega$. Clearly, finiteness of the classical Vapnik-Chervonenkis dimension of $\mathscr C$ is a sufficient, but no longer necessary, condition. Besides, learnability of $\mathscr C$ under non-atomic measures does not imply the uniform Glivenko-Cantelli property with regard to non-atomic measures. Our learnability criterion is stated in terms of a combinatorial parameter $\VC({\mathscr C}\,{\mathrm{mod}}\,\omega_1)$ which we call the VC dimension of $\mathscr C$ modulo countable sets. The new parameter is obtained by ``thickening up'' single points in the definition of VC dimension to uncountable ``clusters''. Equivalently, $\VC(\mathscr C\modd\omega_1)\leq d$ if and only if every countable subclass of $\mathscr C$ has VC dimension $\leq d$ outside a countable subset of $\Omega$. The new parameter can be also expressed as the classical VC dimension of $\mathscr C$ calculated on a suitable subset of a compactification of $\Omega$. We do not make any measurability assumptions on $\mathscr C$, assuming instead the validity of Martin's Axiom (MA).
No associations
LandOfFree
PAC learnability of a concept class under non-atomic measures: a problem by Vidyasagar does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with PAC learnability of a concept class under non-atomic measures: a problem by Vidyasagar, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and PAC learnability of a concept class under non-atomic measures: a problem by Vidyasagar will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-525537