1,106 research outputs found
Random processes via the combinatorial dimension: introductory notes
This is an informal discussion on one of the basic problems in the theory of
empirical processes, addressed in our preprint "Combinatorics of random
processes and sections of convex bodies", which is available at ArXiV and from
our web pages.Comment: 4 page
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
Structural Return Maximization for Reinforcement Learning
Batch Reinforcement Learning (RL) algorithms attempt to choose a policy from
a designer-provided class of policies given a fixed set of training data.
Choosing the policy which maximizes an estimate of return often leads to
over-fitting when only limited data is available, due to the size of the policy
class in relation to the amount of data available. In this work, we focus on
learning policy classes that are appropriately sized to the amount of data
available. We accomplish this by using the principle of Structural Risk
Minimization, from Statistical Learning Theory, which uses Rademacher
complexity to identify a policy class that maximizes a bound on the return of
the best policy in the chosen policy class, given the available data. Unlike
similar batch RL approaches, our bound on return requires only extremely weak
assumptions on the true system
Uniform convergence of Vapnik--Chervonenkis classes under ergodic sampling
We show that if is a complete separable metric space and
is a countable family of Borel subsets of with
finite VC dimension, then, for every stationary ergodic process with values in
, the relative frequencies of sets converge
uniformly to their limiting probabilities. Beyond ergodicity, no assumptions
are imposed on the sampling process, and no regularity conditions are imposed
on the elements of . The result extends existing work of Vapnik
and Chervonenkis, among others, who have studied uniform convergence for i.i.d.
and strongly mixing processes. Our method of proof is new and direct: it does
not rely on symmetrization techniques, probability inequalities or mixing
conditions. The uniform convergence of relative frequencies for VC-major and
VC-graph classes of functions under ergodic sampling is established as a
corollary of the basic result for sets.Comment: Published in at http://dx.doi.org/10.1214/09-AOP511 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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