8,057 research outputs found
Application of a primal-dual interior point algorithm using exact second order information with a novel non-monotone line search method to generally constrained minimax optimization problems
This work presents the application of a primal-dual interior point method to minimax optimisation problems. The algorithm differs significantly from previous approaches as it involves a novel non-monotone line search procedure,
which is based on the use of standard penalty methods as the merit function used for line search. The crucial novel concept is the discretisation of the penalty parameter used over a finite range of orders of magnitude and the
provision of a memory list for each such order. An implementation within a logarithmic barrier algorithm for bounds handling is presented with capabilities for large scale application. Case studies presented demonstrate the capabilities of the proposed methodology, which relies on the reformulation of minimax models into standard nonlinear optimisation models. Some previously reported case studies from the open literature have been solved, and with significantly better optimal solutions identified. We
believe that the nature of the non-monotone line search scheme allows the search procedure to escape from local
minima, hence the encouraging results obtained
Penalized contrast estimator for adaptive density deconvolution
The authors consider the problem of estimating the density of independent
and identically distributed variables , from a sample
where , , is a noise
independent of , with having known distribution. They
present a model selection procedure allowing to construct an adaptive estimator
of and to find non-asymptotic bounds for its
-risk. The estimator achieves the minimax rate of
convergence, in most cases where lowers bounds are available. A simulation
study gives an illustration of the good practical performances of the method
A simple forward selection procedure based on false discovery rate control
We propose the use of a new false discovery rate (FDR) controlling procedure
as a model selection penalized method, and compare its performance to that of
other penalized methods over a wide range of realistic settings: nonorthogonal
design matrices, moderate and large pool of explanatory variables, and both
sparse and nonsparse models, in the sense that they may include a small and
large fraction of the potential variables (and even all). The comparison is
done by a comprehensive simulation study, using a quantitative framework for
performance comparisons in the form of empirical minimaxity relative to a
"random oracle": the oracle model selection performance on data dependent
forward selected family of potential models. We show that FDR based procedures
have good performance, and in particular the newly proposed method, emerges as
having empirical minimax performance. Interestingly, using FDR level of 0.05 is
a global best.Comment: Published in at http://dx.doi.org/10.1214/08-AOAS194 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Adaptive estimation of covariance matrices via Cholesky decomposition
This paper studies the estimation of a large covariance matrix. We introduce
a novel procedure called ChoSelect based on the Cholesky factor of the inverse
covariance. This method uses a dimension reduction strategy by selecting the
pattern of zero of the Cholesky factor. Alternatively, ChoSelect can be
interpreted as a graph estimation procedure for directed Gaussian graphical
models. Our approach is particularly relevant when the variables under study
have a natural ordering (e.g. time series) or more generally when the Cholesky
factor is approximately sparse. ChoSelect achieves non-asymptotic oracle
inequalities with respect to the Kullback-Leibler entropy. Moreover, it
satisfies various adaptive properties from a minimax point of view. We also
introduce and study a two-stage procedure that combines ChoSelect with the
Lasso. This last method enables the practitioner to choose his own trade-off
between statistical efficiency and computational complexity. Moreover, it is
consistent under weaker assumptions than the Lasso. The practical performances
of the different procedures are assessed on numerical examples
Statistical properties of the method of regularization with periodic Gaussian reproducing kernel
The method of regularization with the Gaussian reproducing kernel is popular
in the machine learning literature and successful in many practical
applications.
In this paper we consider the periodic version of the Gaussian kernel
regularization.
We show in the white noise model setting, that in function spaces of very
smooth functions, such as the infinite-order Sobolev space and the space of
analytic functions, the method under consideration is asymptotically minimax;
in finite-order Sobolev spaces, the method is rate optimal, and the efficiency
in terms of constant when compared with the minimax estimator is reasonably
high. The smoothing parameters in the periodic Gaussian regularization can be
chosen adaptively without loss of asymptotic efficiency. The results derived in
this paper give a partial explanation of the success of the
Gaussian reproducing kernel in practice. Simulations are carried out to study
the finite sample properties of the periodic Gaussian regularization.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Statistics
(http://www.imstat.org/aos/) at http://dx.doi.org/10.1214/00905360400000045
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