1,414 research outputs found
Rates of convergence of rho-estimators for sets of densities satisfying shape constraints
The purpose of this paper is to pursue our study of rho-estimators built from
i.i.d. observations that we defined in Baraud et al. (2014). For a
\rho-estimator based on some model S (which means that the estimator belongs to
S) and a true distribution of the observations that also belongs to S, the risk
(with squared Hellinger loss) is bounded by a quantity which can be viewed as a
dimension function of the model and is often related to the "metric dimension"
of this model, as defined in Birg\'e (2006). This is a minimax point of view
and it is well-known that it is pessimistic. Typically, the bound is accurate
for most points in the model but may be very pessimistic when the true
distribution belongs to some specific part of it. This is the situation that we
want to investigate here. For some models, like the set of decreasing densities
on [0,1], there exist specific points in the model that we shall call
"extremal" and for which the risk is substantially smaller than the typical
risk. Moreover, the risk at a non-extremal point of the model can be bounded by
the sum of the risk bound at a well-chosen extremal point plus the square of
its distance to this point. This implies that if the true density is close
enough to an extremal point, the risk at this point may be smaller than the
minimax risk on the model and this actually remains true even if the true
density does not belong to the model. The result is based on some refined
bounds on the suprema of empirical processes that are established in Baraud
(2016).Comment: 24 page
Caveats for information bottleneck in deterministic scenarios
Information bottleneck (IB) is a method for extracting information from one
random variable that is relevant for predicting another random variable
. To do so, IB identifies an intermediate "bottleneck" variable that has
low mutual information and high mutual information . The "IB
curve" characterizes the set of bottleneck variables that achieve maximal
for a given , and is typically explored by maximizing the "IB
Lagrangian", . In some cases, is a deterministic
function of , including many classification problems in supervised learning
where the output class is a deterministic function of the input . We
demonstrate three caveats when using IB in any situation where is a
deterministic function of : (1) the IB curve cannot be recovered by
maximizing the IB Lagrangian for different values of ; (2) there are
"uninteresting" trivial solutions at all points of the IB curve; and (3) for
multi-layer classifiers that achieve low prediction error, different layers
cannot exhibit a strict trade-off between compression and prediction, contrary
to a recent proposal. We also show that when is a small perturbation away
from being a deterministic function of , these three caveats arise in an
approximate way. To address problem (1), we propose a functional that, unlike
the IB Lagrangian, can recover the IB curve in all cases. We demonstrate the
three caveats on the MNIST dataset
Slope heuristics and V-Fold model selection in heteroscedastic regression using strongly localized bases
We investigate the optimality for model selection of the so-called slope
heuristics, -fold cross-validation and -fold penalization in a
heteroscedastic with random design regression context. We consider a new class
of linear models that we call strongly localized bases and that generalize
histograms, piecewise polynomials and compactly supported wavelets. We derive
sharp oracle inequalities that prove the asymptotic optimality of the slope
heuristics---when the optimal penalty shape is known---and -fold
penalization. Furthermore, -fold cross-validation seems to be suboptimal for
a fixed value of since it recovers asymptotically the oracle learned from a
sample size equal to of the original amount of data. Our results are
based on genuine concentration inequalities for the true and empirical excess
risks that are of independent interest. We show in our experiments the good
behavior of the slope heuristics for the selection of linear wavelet models.
Furthermore, -fold cross-validation and -fold penalization have
comparable efficiency
Estimating composite functions by model selection
We consider the problem of estimating a function on for
large values of by looking for some best approximation by composite
functions of the form . Our solution is based on model selection and
leads to a very general approach to solve this problem with respect to many
different types of functions and statistical frameworks. In particular,
we handle the problems of approximating by additive functions, single and
multiple index models, neural networks, mixtures of Gaussian densities (when
is a density) among other examples. We also investigate the situation where
for functions and belonging to possibly anisotropic
smoothness classes. In this case, our approach leads to a completely adaptive
estimator with respect to the regularity of .Comment: 37 page
- …