VC Dimension
A fundamental combinatorial quantity in statistical learning theory that characterizes the complexity of a hypothesis space by measuring the size of the largest point set that the hypothesis class can shatter. It serves as a foundational tool for analyzing sample complexity in PAC learning theory.
Timeline (last 90 days)
The paper derives generalization error upper bounds separately for the preliminary estimator and the constrained estimator; these bounds are expressed in terms of complexity measures such as the covering number or VC dimension of the family of measurable sets, and do not depend on the specific structure of the response space.
An arXiv paper (arXiv:2610.10600) proves that a learning algorithm can be constructed whose evaluated conditional mutual information (eCMI) reaches O(d) order in the realizable case, thereby recovering the optimal PAC guarantees for VC classes.
The classical theorem by Blumer et al. proves that in the realizable case, sample complexity is linearly related to VC dimension d, i.e., samples on the order of Θ(d/ε) are sufficient to guarantee a generalization error of ε.
The Vapnik-Chervonenkis (VC) dimension is the core combinatorial quantity characterizing the complexity of a hypothesis space, measuring the maximum size of a point set that the hypothesis class can shatter.
The independent and identically distributed (i.i.d.) assumption is the cornerstone of classical statistical learning theory and is a prerequisite for the PAC learning framework and VC dimension theory
The theoretical foundations of empirical risk minimization can be traced back to the work of Vapnik and Chervonenkis in the 1970s, who proved that under conditions such as finite VC dimension of the hypothesis space, empirical risk minimizers can converge to true risk minimizers at a controllable rate.
All Facts (6)
The paper derives generalization error upper bounds separately for the preliminary estimator and the constrained estimator; these bounds are expressed in terms of complexity measures such as the covering number or VC dimension of the family of measurable sets, and do not depend on the specific structure of the response space.
50%UnverifiedAn arXiv paper (arXiv:2610.10600) proves that a learning algorithm can be constructed whose evaluated conditional mutual information (eCMI) reaches O(d) order in the realizable case, thereby recovering the optimal PAC guarantees for VC classes.
50%UnverifiedThe classical theorem by Blumer et al. proves that in the realizable case, sample complexity is linearly related to VC dimension d, i.e., samples on the order of Θ(d/ε) are sufficient to guarantee a generalization error of ε.
50%UnverifiedThe Vapnik-Chervonenkis (VC) dimension is the core combinatorial quantity characterizing the complexity of a hypothesis space, measuring the maximum size of a point set that the hypothesis class can shatter.
50%UnverifiedThe independent and identically distributed (i.i.d.) assumption is the cornerstone of classical statistical learning theory and is a prerequisite for the PAC learning framework and VC dimension theory
50%UnverifiedThe theoretical foundations of empirical risk minimization can be traced back to the work of Vapnik and Chervonenkis in the 1970s, who proved that under conditions such as finite VC dimension of the hypothesis space, empirical risk minimizers can converge to true risk minimizers at a controllable rate.
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