1,858 research outputs found
Continuous and Random Vapnik-Chervonenkis Classes
We show that if is a dependent theory then so is its Keisler
randomisation . In order to do this we generalise the notion of a
Vapnik-Chervonenkis class to families of -valued functions (a
\emph{continuous} Vapnik-Chervonenkis class), and we characterise families of
functions having this property via the growth rate of the mean width of an
associated family of convex compacts
Bounding Embeddings of VC Classes into Maximum Classes
One of the earliest conjectures in computational learning theory-the Sample
Compression conjecture-asserts that concept classes (equivalently set systems)
admit compression schemes of size linear in their VC dimension. To-date this
statement is known to be true for maximum classes---those that possess maximum
cardinality for their VC dimension. The most promising approach to positively
resolving the conjecture is by embedding general VC classes into maximum
classes without super-linear increase to their VC dimensions, as such
embeddings would extend the known compression schemes to all VC classes. We
show that maximum classes can be characterised by a local-connectivity property
of the graph obtained by viewing the class as a cubical complex. This geometric
characterisation of maximum VC classes is applied to prove a negative embedding
result which demonstrates VC-d classes that cannot be embedded in any maximum
class of VC dimension lower than 2d. On the other hand, we show that every VC-d
class C embeds in a VC-(d+D) maximum class where D is the deficiency of C,
i.e., the difference between the cardinalities of a maximum VC-d class and of
C. For VC-2 classes in binary n-cubes for 4 <= n <= 6, we give best possible
results on embedding into maximum classes. For some special classes of Boolean
functions, relationships with maximum classes are investigated. Finally we give
a general recursive procedure for embedding VC-d classes into VC-(d+k) maximum
classes for smallest k.Comment: 22 pages, 2 figure
An equivalence result for VC classes of sets
Let R and θ be infinite sets and let A # R × θ. We show that the class of projections of A onto R is a Vapnik–Chervonenkis (VC) class of sets if and only if the class of projections of A onto θ is a VC class. We illustrate the result in the context of semiparametric estimation of a transformation model. In this application, the VC property is hard to establish for the projection class of interest but easy to establish for the other projection class
Fast DD-classification of functional data
A fast nonparametric procedure for classifying functional data is introduced.
It consists of a two-step transformation of the original data plus a classifier
operating on a low-dimensional hypercube. The functional data are first mapped
into a finite-dimensional location-slope space and then transformed by a
multivariate depth function into the -plot, which is a subset of the unit
hypercube. This transformation yields a new notion of depth for functional
data. Three alternative depth functions are employed for this, as well as two
rules for the final classification on . The resulting classifier has
to be cross-validated over a small range of parameters only, which is
restricted by a Vapnik-Cervonenkis bound. The entire methodology does not
involve smoothing techniques, is completely nonparametric and allows to achieve
Bayes optimality under standard distributional settings. It is robust,
efficiently computable, and has been implemented in an R environment.
Applicability of the new approach is demonstrated by simulations as well as a
benchmark study
Fast rates in statistical and online learning
The speed with which a learning algorithm converges as it is presented with
more data is a central problem in machine learning --- a fast rate of
convergence means less data is needed for the same level of performance. The
pursuit of fast rates in online and statistical learning has led to the
discovery of many conditions in learning theory under which fast learning is
possible. We show that most of these conditions are special cases of a single,
unifying condition, that comes in two forms: the central condition for 'proper'
learning algorithms that always output a hypothesis in the given model, and
stochastic mixability for online algorithms that may make predictions outside
of the model. We show that under surprisingly weak assumptions both conditions
are, in a certain sense, equivalent. The central condition has a
re-interpretation in terms of convexity of a set of pseudoprobabilities,
linking it to density estimation under misspecification. For bounded losses, we
show how the central condition enables a direct proof of fast rates and we
prove its equivalence to the Bernstein condition, itself a generalization of
the Tsybakov margin condition, both of which have played a central role in
obtaining fast rates in statistical learning. Yet, while the Bernstein
condition is two-sided, the central condition is one-sided, making it more
suitable to deal with unbounded losses. In its stochastic mixability form, our
condition generalizes both a stochastic exp-concavity condition identified by
Juditsky, Rigollet and Tsybakov and Vovk's notion of mixability. Our unifying
conditions thus provide a substantial step towards a characterization of fast
rates in statistical learning, similar to how classical mixability
characterizes constant regret in the sequential prediction with expert advice
setting.Comment: 69 pages, 3 figure
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