4,024 research outputs found
Does generalization performance of regularization learning depend on ? A negative example
-regularization has been demonstrated to be an attractive technique in
machine learning and statistical modeling. It attempts to improve the
generalization (prediction) capability of a machine (model) through
appropriately shrinking its coefficients. The shape of a estimator
differs in varying choices of the regularization order . In particular,
leads to the LASSO estimate, while corresponds to the smooth
ridge regression. This makes the order a potential tuning parameter in
applications. To facilitate the use of -regularization, we intend to
seek for a modeling strategy where an elaborative selection on is
avoidable. In this spirit, we place our investigation within a general
framework of -regularized kernel learning under a sample dependent
hypothesis space (SDHS). For a designated class of kernel functions, we show
that all estimators for attain similar generalization
error bounds. These estimated bounds are almost optimal in the sense that up to
a logarithmic factor, the upper and lower bounds are asymptotically identical.
This finding tentatively reveals that, in some modeling contexts, the choice of
might not have a strong impact in terms of the generalization capability.
From this perspective, can be arbitrarily specified, or specified merely by
other no generalization criteria like smoothness, computational complexity,
sparsity, etc..Comment: 35 pages, 3 figure
Variable selection in semiparametric regression modeling
In this paper, we are concerned with how to select significant variables in
semiparametric modeling. Variable selection for semiparametric regression
models consists of two components: model selection for nonparametric components
and selection of significant variables for the parametric portion. Thus,
semiparametric variable selection is much more challenging than parametric
variable selection (e.g., linear and generalized linear models) because
traditional variable selection procedures including stepwise regression and the
best subset selection now require separate model selection for the
nonparametric components for each submodel. This leads to a very heavy
computational burden. In this paper, we propose a class of variable selection
procedures for semiparametric regression models using nonconcave penalized
likelihood. We establish the rate of convergence of the resulting estimate.
With proper choices of penalty functions and regularization parameters, we show
the asymptotic normality of the resulting estimate and further demonstrate that
the proposed procedures perform as well as an oracle procedure. A
semiparametric generalized likelihood ratio test is proposed to select
significant variables in the nonparametric component. We investigate the
asymptotic behavior of the proposed test and demonstrate that its limiting null
distribution follows a chi-square distribution which is independent of the
nuisance parameters. Extensive Monte Carlo simulation studies are conducted to
examine the finite sample performance of the proposed variable selection
procedures.Comment: Published in at http://dx.doi.org/10.1214/009053607000000604 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Model selection of polynomial kernel regression
Polynomial kernel regression is one of the standard and state-of-the-art
learning strategies. However, as is well known, the choices of the degree of
polynomial kernel and the regularization parameter are still open in the realm
of model selection. The first aim of this paper is to develop a strategy to
select these parameters. On one hand, based on the worst-case learning rate
analysis, we show that the regularization term in polynomial kernel regression
is not necessary. In other words, the regularization parameter can decrease
arbitrarily fast when the degree of the polynomial kernel is suitable tuned. On
the other hand,taking account of the implementation of the algorithm, the
regularization term is required. Summarily, the effect of the regularization
term in polynomial kernel regression is only to circumvent the " ill-condition"
of the kernel matrix. Based on this, the second purpose of this paper is to
propose a new model selection strategy, and then design an efficient learning
algorithm. Both theoretical and experimental analysis show that the new
strategy outperforms the previous one. Theoretically, we prove that the new
learning strategy is almost optimal if the regression function is smooth.
Experimentally, it is shown that the new strategy can significantly reduce the
computational burden without loss of generalization capability.Comment: 29 pages, 4 figure
Regularized Regression Problem in hyper-RKHS for Learning Kernels
This paper generalizes the two-stage kernel learning framework, illustrates
its utility for kernel learning and out-of-sample extensions, and proves
{asymptotic} convergence results for the introduced kernel learning model.
Algorithmically, we extend target alignment by hyper-kernels in the two-stage
kernel learning framework. The associated kernel learning task is formulated as
a regression problem in a hyper-reproducing kernel Hilbert space (hyper-RKHS),
i.e., learning on the space of kernels itself. To solve this problem, we
present two regression models with bivariate forms in this space, including
kernel ridge regression (KRR) and support vector regression (SVR) in the
hyper-RKHS. By doing so, it provides significant model flexibility for kernel
learning with outstanding performance in real-world applications. Specifically,
our kernel learning framework is general, that is, the learned underlying
kernel can be positive definite or indefinite, which adapts to various
requirements in kernel learning. Theoretically, we study the convergence
behavior of these learning algorithms in the hyper-RKHS and derive the learning
rates. Different from the traditional approximation analysis in RKHS, our
analyses need to consider the non-trivial independence of pairwise samples and
the characterisation of hyper-RKHS. To the best of our knowledge, this is the
first work in learning theory to study the approximation performance of
regularized regression problem in hyper-RKHS.Comment: 25 pages, 3 figure
Domain Adaptation: Learning Bounds and Algorithms
This paper addresses the general problem of domain adaptation which arises in
a variety of applications where the distribution of the labeled sample
available somewhat differs from that of the test data. Building on previous
work by Ben-David et al. (2007), we introduce a novel distance between
distributions, discrepancy distance, that is tailored to adaptation problems
with arbitrary loss functions. We give Rademacher complexity bounds for
estimating the discrepancy distance from finite samples for different loss
functions. Using this distance, we derive novel generalization bounds for
domain adaptation for a wide family of loss functions. We also present a series
of novel adaptation bounds for large classes of regularization-based
algorithms, including support vector machines and kernel ridge regression based
on the empirical discrepancy. This motivates our analysis of the problem of
minimizing the empirical discrepancy for various loss functions for which we
also give novel algorithms. We report the results of preliminary experiments
that demonstrate the benefits of our discrepancy minimization algorithms for
domain adaptation.Comment: 12 pages, 4 figure
Non-convex regularization in remote sensing
In this paper, we study the effect of different regularizers and their
implications in high dimensional image classification and sparse linear
unmixing. Although kernelization or sparse methods are globally accepted
solutions for processing data in high dimensions, we present here a study on
the impact of the form of regularization used and its parametrization. We
consider regularization via traditional squared (2) and sparsity-promoting (1)
norms, as well as more unconventional nonconvex regularizers (p and Log Sum
Penalty). We compare their properties and advantages on several classification
and linear unmixing tasks and provide advices on the choice of the best
regularizer for the problem at hand. Finally, we also provide a fully
functional toolbox for the community.Comment: 11 pages, 11 figure
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