291,993 research outputs found
Learning Geometric Concepts with Nasty Noise
We study the efficient learnability of geometric concept classes -
specifically, low-degree polynomial threshold functions (PTFs) and
intersections of halfspaces - when a fraction of the data is adversarially
corrupted. We give the first polynomial-time PAC learning algorithms for these
concept classes with dimension-independent error guarantees in the presence of
nasty noise under the Gaussian distribution. In the nasty noise model, an
omniscient adversary can arbitrarily corrupt a small fraction of both the
unlabeled data points and their labels. This model generalizes well-studied
noise models, including the malicious noise model and the agnostic (adversarial
label noise) model. Prior to our work, the only concept class for which
efficient malicious learning algorithms were known was the class of
origin-centered halfspaces.
Specifically, our robust learning algorithm for low-degree PTFs succeeds
under a number of tame distributions -- including the Gaussian distribution
and, more generally, any log-concave distribution with (approximately) known
low-degree moments. For LTFs under the Gaussian distribution, we give a
polynomial-time algorithm that achieves error , where
is the noise rate. At the core of our PAC learning results is an efficient
algorithm to approximate the low-degree Chow-parameters of any bounded function
in the presence of nasty noise. To achieve this, we employ an iterative
spectral method for outlier detection and removal, inspired by recent work in
robust unsupervised learning. Our aforementioned algorithm succeeds for a range
of distributions satisfying mild concentration bounds and moment assumptions.
The correctness of our robust learning algorithm for intersections of
halfspaces makes essential use of a novel robust inverse independence lemma
that may be of broader interest
Hybrid Shrinkage Estimators Using Penalty Bases For The Ordinal One-Way Layout
This paper constructs improved estimators of the means in the Gaussian
saturated one-way layout with an ordinal factor. The least squares estimator
for the mean vector in this saturated model is usually inadmissible. The hybrid
shrinkage estimators of this paper exploit the possibility of slow variation in
the dependence of the means on the ordered factor levels but do not assume it
and respond well to faster variation if present. To motivate the development,
candidate penalized least squares (PLS) estimators for the mean vector of a
one-way layout are represented as shrinkage estimators relative to the penalty
basis for the regression space. This canonical representation suggests further
classes of candidate estimators for the unknown means: monotone shrinkage (MS)
estimators or soft-thresholding (ST) estimators or, most generally, hybrid
shrinkage (HS) estimators that combine the preceding two strategies. Adaptation
selects the estimator within a candidate class that minimizes estimated risk.
Under the Gaussian saturated one-way layout model, such adaptive estimators
minimize risk asymptotically over the class of candidate estimators as the
number of factor levels tends to infinity. Thereby, adaptive HS estimators
asymptotically dominate adaptive MS and adaptive ST estimators as well as the
least squares estimator. Local annihilators of polynomials, among them
difference operators, generate penalty bases suitable for a range of numerical
examples.Comment: Published at http://dx.doi.org/10.1214/009053604000000652 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Depth weighted scatter estimators
General depth weighted scatter estimators are introduced and investigated.
For general depth functions, we find out that these affine equivariant scatter
estimators are Fisher consistent and unbiased for a wide range of multivariate
distributions, and show that the sample scatter estimators are strong and
\sqrtn-consistent and asymptotically normal, and the influence functions of the
estimators exist and are bounded in general. We then concentrate on a specific
case of the general depth weighted scatter estimators, the projection depth
weighted scatter estimators, which include as a special case the well-known
Stahel-Donoho scatter estimator whose limiting distribution has long been open
until this paper. Large sample behavior, including consistency and asymptotic
normality, and efficiency and finite sample behavior, including breakdown point
and relative efficiency of the sample projection depth weighted scatter
estimators, are thoroughly investigated. The influence function and the maximum
bias of the projection depth weighted scatter estimators are derived and
examined. Unlike typical high-breakdown competitors, the projection depth
weighted scatter estimators can integrate high breakdown point and high
efficiency while enjoying a bounded-influence function and a moderate maximum
bias curve. Comparisons with leading estimators on asymptotic relative
efficiency and gross error sensitivity reveal that the projection depth
weighted scatter estimators behave very well overall and, consequently,
represent very favorable choices of affine equivariant multivariate scatter
estimators.Comment: Published at http://dx.doi.org/10.1214/009053604000000922 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Symmetry, regression design, and sampling distributions
When values of regressors are symmetrically disposed, many M-estimators in a wide class of models have a reflection property, namely, that as the signs of the coefficients on regressors are reversed, their estimators' sampling distribution is reflected about the origin. When the coefficients are zero, sign reversal can have no effect. So in this case, the sampling distribution of regression coefficient estimators is symmetric about zero, the estimators are median unbiased and, when moments exist, the estimators are exactly uncorrelated with estimators of other parameters. The result is unusual in that it does not require response variates to have symmetric conditional distributions. It demonstrates the potential importance of covariate design in determining the distributions of estimators, and it is useful in designing and interpreting Monte Carlo experiments. The result is illustrated by a Monte Carlo experiment in which maximum likelihood and symmetrically censored least-squares estimators are calculated for small samples from a censored normal linear regression, Tobit, model. © 1994, Cambridge University Press. All rights reserved
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