21 research outputs found
Affine Invariant Covariance Estimation for Heavy-Tailed Distributions
In this work we provide an estimator for the covariance matrix of a
heavy-tailed multivariate distributionWe prove that the proposed estimator
admits an \textit{affine-invariant} bound of the form
in high probability, where is the
unknown covariance matrix, and is the positive semidefinite
order on symmetric matrices. The result only requires the existence of
fourth-order moments, and allows for where is a measure of kurtosis of the
distribution, is the dimensionality of the space, is the sample size,
and is the desired confidence level. More generally, we can allow
for regularization with level , then gets replaced with the
degrees of freedom number. Denoting the condition
number of , the computational cost of the novel estimator is , which is comparable to the cost of the
sample covariance estimator in the statistically interesing regime .
We consider applications of our estimator to eigenvalue estimation with
relative error, and to ridge regression with heavy-tailed random design
Finite-sample Analysis of M-estimators using Self-concordance
We demonstrate how self-concordance of the loss can be exploited to obtain
asymptotically optimal rates for M-estimators in finite-sample regimes. We
consider two classes of losses: (i) canonically self-concordant losses in the
sense of Nesterov and Nemirovski (1994), i.e., with the third derivative
bounded with the power of the second; (ii) pseudo self-concordant losses,
for which the power is removed, as introduced by Bach (2010). These classes
contain some losses arising in generalized linear models, including logistic
regression; in addition, the second class includes some common pseudo-Huber
losses. Our results consist in establishing the critical sample size sufficient
to reach the asymptotically optimal excess risk for both classes of losses.
Denoting the parameter dimension, and the effective
dimension which takes into account possible model misspecification, we find the
critical sample size to be for canonically
self-concordant losses, and for pseudo
self-concordant losses, where is the problem-dependent local curvature
parameter. In contrast to the existing results, we only impose local
assumptions on the data distribution, assuming that the calibrated design,
i.e., the design scaled with the square root of the second derivative of the
loss, is subgaussian at the best predictor . Moreover, we obtain the
improved bounds on the critical sample size, scaling near-linearly in
, under the extra assumption that the calibrated design
is subgaussian in the Dikin ellipsoid of . Motivated by these
findings, we construct canonically self-concordant analogues of the Huber and
logistic losses with improved statistical properties. Finally, we extend some
of these results to -regularized M-estimators in high dimensions
Affine Invariant Covariance Estimation for Heavy-Tailed Distributions
International audienceIn this work we provide an estimator for the covariance matrix of a heavy-tailed multivariate distributionWe prove that the proposed estimator admits an \textit{affine-invariant} bound of the form in high probability, where is the unknown covariance matrix, and is the positive semidefinite order on symmetric matrices. The result only requires the existence of fourth-order moments, and allows for where is a measure of kurtosis of the distribution, is the dimensionality of the space, is the sample size, and is the desired confidence level. More generally, we can allow for regularization with level , then gets replaced with the degrees of freedom number. Denoting the condition number of , the computational cost of the novel estimator is , which is comparable to the cost of the sample covariance estimator in the statistically interesing regime . We consider applications of our estimator to eigenvalue estimation with relative error, and to ridge regression with heavy-tailed random design
Finite-sample analysis of M-estimators using self-concordance
The classical asymptotic theory for parametric -estimators guarantees that, in the limit of infinite sample size, the excess risk has a chi-square type distribution, even in the misspecified case. We demonstrate how self-concordance of the loss allows to characterize the critical sample size sufficient to guarantee a chi-square type in-probability bound for the excess risk. Specifically, we consider two classes of losses: (i) self-concordant losses in the classical sense of Nesterov and Nemirovski, i.e., whose third derivative is uniformly bounded with the power of the second derivative; (ii) pseudo self-concordant losses, for which the power is removed. These classes contain losses corresponding to several generalized linear models, including the logistic loss and pseudo-Huber losses. Our basic result under minimal assumptions bounds the critical sample size by where the parameter dimension and the effective dimension that accounts for model misspecification. In contrast to the existing results, we only impose local assumptions that concern the population risk minimizer . Namely, we assume that the calibrated design, i.e., design scaled by the square root of the second derivative of the loss, is subgaussian at . Besides, for type-ii losses we require boundedness of a certain measure of curvature of the population risk at .Our improved result bounds the critical sample size from above as under slightly stronger assumptions. Namely, the local assumptions must hold in the neighborhood of given by the Dikin ellipsoid of the population risk. Interestingly, we find that, for logistic regression with Gaussian design, there is no actual restriction of conditions: the subgaussian parameter and curvature measure remain near-constant over the Dikin ellipsoid. Finally, we extend some of these results to -penalized estimators in high dimensions
Adaptive Denoising of Signals with Shift-Invariant Structure
We study the problem of discrete-time signal denoising, following the line of
research initiated by [Nem91] and further developed in [JN09, JN10, HJNO15,
OHJN16]. Previous papers considered the following setup: the signal is assumed
to admit a convolution-type linear oracle -- an unknown linear estimator in the
form of the convolution of the observations with an unknown time-invariant
filter with small -norm. It was shown that such an oracle can be
"mimicked" by an efficiently computable non-linear convolution-type estimator,
in which the filter minimizes the Fourier-domain -norm of the
residual, regularized by the Fourier-domain -norm of the filter.
Following [OHJN16], here we study an alternative family of estimators,
replacing the -norm of the residual with the -norm. Such
estimators are found to have better statistical properties, in particular, we
prove sharp oracle inequalities for their -loss. Our guarantees require
an extra assumption of approximate shift-invariance: the signal must be
-close, in -metric, to some shift-invariant linear subspace
with bounded dimension . However, this subspace can be completely unknown,
and the remainder terms in the oracle inequalities scale at most polynomially
with and . In conclusion, we show that the new assumption
implies the previously considered one, providing explicit constructions of the
convolution-type linear oracles with -norm bounded in terms of
parameters and