Shifting, One-Inclusion Mistake Bounds and Tight Multiclass Expected Risk Bounds

Formats Non-Member Member
$15 $15
Learn how you can qualify for the best price for this item!
Become an IEEE Member or Subscribe to
IEEE Xplore for exclusive pricing!
close button

puzzle piece

IEEE membership options for an individual and IEEE Xplore subscriptions for an organization offer the most affordable access to essential journal articles, conference papers, standards, books, and eLearning courses.

Learn more about:

IEEE membership

IEEE Xplore subscriptions

3 Author(s)

Under the prediction model of learning, a prediction strategy is presented with an i.i.d. sample of n — 1 points in X and corresponding labels from a concept ƒ ∈ F, and aims to minimize the worst-case probability of erring on an nth point. By exploiting the structure of F, Haussler et al. achieved a VC(F)/n bound for the natural one-inclusion prediction strategy, improving on bounds implied by PAC-type results by a O(log n) factor. The key data structure in their result is the natural subgraph of the hypercube.the one-inclusion graph; the key step is a d = VC(F) bound on one-inclusion graph density. The first main result of this paper is a density bound of n (n - 1/≤d - 1 / (n/≤d) < d, which positively resolves a conjecture of Kuzmin & Warmuth relating to their unlabeled Peeling compression scheme and also leads to an improved mistake bound for the randomized (deterministic) one-inclusion strategy for all d (for d ≈ Θ(n)). The proof uses a new form of VC-invariant shifting and a group-theoretic symmetrization. Our second main result is a k-class analogue of the d/n mistake bound, replacing the VC-dimension by the Pollard pseudo-dimension and the one-inclusion strategy by its natural hypergraph generalization. This bound on expected risk improves on known PAC-based results by a factor of O(log n) and is shown to be optimal up to a O(log k) factor. The combinatorial technique of shifting takes a central role in understanding the one-inclusion (hyper)graph and is a running theme throughout.