13,085 research outputs found
Approximate resilience, monotonicity, and the complexity of agnostic learning
A function is -resilient if all its Fourier coefficients of degree at
most are zero, i.e., is uncorrelated with all low-degree parities. We
study the notion of of Boolean
functions, where we say that is -approximately -resilient if
is -close to a -valued -resilient function in
distance. We show that approximate resilience essentially characterizes the
complexity of agnostic learning of a concept class over the uniform
distribution. Roughly speaking, if all functions in a class are far from
being -resilient then can be learned agnostically in time and
conversely, if contains a function close to being -resilient then
agnostic learning of in the statistical query (SQ) framework of Kearns has
complexity of at least . This characterization is based on the
duality between approximation by degree- polynomials and
approximate -resilience that we establish. In particular, it implies that
approximation by low-degree polynomials, known to be sufficient for
agnostic learning over product distributions, is in fact necessary.
Focusing on monotone Boolean functions, we exhibit the existence of
near-optimal -approximately
-resilient monotone functions for all
. Prior to our work, it was conceivable even that every monotone
function is -far from any -resilient function. Furthermore, we
construct simple, explicit monotone functions based on and that are close to highly resilient functions. Our constructions are
based on a fairly general resilience analysis and amplification. These
structural results, together with the characterization, imply nearly optimal
lower bounds for agnostic learning of monotone juntas
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
Adaptive Online Sequential ELM for Concept Drift Tackling
A machine learning method needs to adapt to over time changes in the
environment. Such changes are known as concept drift. In this paper, we propose
concept drift tackling method as an enhancement of Online Sequential Extreme
Learning Machine (OS-ELM) and Constructive Enhancement OS-ELM (CEOS-ELM) by
adding adaptive capability for classification and regression problem. The
scheme is named as adaptive OS-ELM (AOS-ELM). It is a single classifier scheme
that works well to handle real drift, virtual drift, and hybrid drift. The
AOS-ELM also works well for sudden drift and recurrent context change type. The
scheme is a simple unified method implemented in simple lines of code. We
evaluated AOS-ELM on regression and classification problem by using concept
drift public data set (SEA and STAGGER) and other public data sets such as
MNIST, USPS, and IDS. Experiments show that our method gives higher kappa value
compared to the multiclassifier ELM ensemble. Even though AOS-ELM in practice
does not need hidden nodes increase, we address some issues related to the
increasing of the hidden nodes such as error condition and rank values. We
propose taking the rank of the pseudoinverse matrix as an indicator parameter
to detect underfitting condition.Comment: Hindawi Publishing. Computational Intelligence and Neuroscience
Volume 2016 (2016), Article ID 8091267, 17 pages Received 29 January 2016,
Accepted 17 May 2016. Special Issue on "Advances in Neural Networks and
Hybrid-Metaheuristics: Theory, Algorithms, and Novel Engineering
Applications". Academic Editor: Stefan Hauf
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