9,192 research outputs found
Maximum Margin Multiclass Nearest Neighbors
We develop a general framework for margin-based multicategory classification
in metric spaces. The basic work-horse is a margin-regularized version of the
nearest-neighbor classifier. We prove generalization bounds that match the
state of the art in sample size and significantly improve the dependence on
the number of classes . Our point of departure is a nearly Bayes-optimal
finite-sample risk bound independent of . Although -free, this bound is
unregularized and non-adaptive, which motivates our main result: Rademacher and
scale-sensitive margin bounds with a logarithmic dependence on . As the best
previous risk estimates in this setting were of order , our bound is
exponentially sharper. From the algorithmic standpoint, in doubling metric
spaces our classifier may be trained on examples in time and
evaluated on new points in time
Scale-sensitive Psi-dimensions: the Capacity Measures for Classifiers Taking Values in R^Q
Bounds on the risk play a crucial role in statistical learning theory. They
usually involve as capacity measure of the model studied the VC dimension or
one of its extensions. In classification, such "VC dimensions" exist for models
taking values in {0, 1}, {1,..., Q} and R. We introduce the generalizations
appropriate for the missing case, the one of models with values in R^Q. This
provides us with a new guaranteed risk for M-SVMs which appears superior to the
existing one
Efficient Classification for Metric Data
Recent advances in large-margin classification of data residing in general
metric spaces (rather than Hilbert spaces) enable classification under various
natural metrics, such as string edit and earthmover distance. A general
framework developed for this purpose by von Luxburg and Bousquet [JMLR, 2004]
left open the questions of computational efficiency and of providing direct
bounds on generalization error.
We design a new algorithm for classification in general metric spaces, whose
runtime and accuracy depend on the doubling dimension of the data points, and
can thus achieve superior classification performance in many common scenarios.
The algorithmic core of our approach is an approximate (rather than exact)
solution to the classical problems of Lipschitz extension and of Nearest
Neighbor Search. The algorithm's generalization performance is guaranteed via
the fat-shattering dimension of Lipschitz classifiers, and we present
experimental evidence of its superiority to some common kernel methods. As a
by-product, we offer a new perspective on the nearest neighbor classifier,
which yields significantly sharper risk asymptotics than the classic analysis
of Cover and Hart [IEEE Trans. Info. Theory, 1967].Comment: This is the full version of an extended abstract that appeared in
Proceedings of the 23rd COLT, 201
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