89 research outputs found
Adapting to Unknown Smoothness by Aggregation of Thresholded Wavelet Estimators
We study the performances of an adaptive procedure based on a convex
combination, with data-driven weights, of term-by-term thresholded wavelet
estimators. For the bounded regression model, with random uniform design, and
the nonparametric density model, we show that the resulting estimator is
optimal in the minimax sense over all Besov balls under the risk, without
any logarithm factor
Sharper lower bounds on the performance of the empirical risk minimization algorithm
We present an argument based on the multidimensional and the uniform central
limit theorems, proving that, under some geometrical assumptions between the
target function and the learning class , the excess risk of the
empirical risk minimization algorithm is lower bounded by
where
is a canonical Gaussian process associated with (a well chosen subset of
) and is a parameter governing the oscillations of the empirical
excess risk function over a small ball in .Comment: Published in at http://dx.doi.org/10.3150/09-BEJ225 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Optimal rates and adaptation in the single-index model using aggregation
We want to recover the regression function in the single-index model. Using
an aggregation algorithm with local polynomial estimators, we answer in
particular to the second part of Question~2 from Stone (1982) on the optimal
convergence rate. The procedure constructed here has strong adaptation
properties: it adapts both to the smoothness of the link function and to the
unknown index. Moreover, the procedure locally adapts to the distribution of
the design. We propose new upper bounds for the local polynomial estimator
(which are results of independent interest) that allows a fairly general
design. The behavior of this algorithm is studied through numerical
simulations. In particular, we show empirically that it improves strongly over
empirical risk minimization.Comment: 36 page
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