arXiv subjects

Ingo Steinwart

Publications and source records attributed to Ingo Steinwart.

53 records · Page 3Linked to original sources

arXiv2016

Kernel Density Estimation for Dynamical Systems

We study the density estimation problem with observations generated by certain dynamical systems that admit a unique underlying invariant Lebesgue density. Observations drawn from dynamical systems are not independent and moreover, usual mixing concepts may not be appropriate for measuring the dependence among these observations. By employing the $\mathcal{C}$-mixing concept to measure the dependence, we conduct statistical analysis on the consistency and convergence of the kernel density estimator. Our main results are as follows: First, we show that with properly chosen bandwidth, the kernel density estimator is universally consistent under $L_1$-norm; Second, we establish convergence rates for the estimator with respect to several classes of dynamical systems under $L_1$-norm. In the analysis, the density function $f$ is only assumed to be Hölder continuous which is a weak assumption in the literature of nonparametric density estimation and also more realistic in the dynamical system context. Last but not least, we prove that the same convergence rates of the estimator under $L_\infty$-norm and $L_1$-norm can be achieved when the density function is Hölder continuous, compactly supported and bounded. The bandwidth selection problem of the kernel density estimator for dynamical system is also discussed in our study via numerical simulations.

stat.ML↗

arXiv2016

A Short Note on the Comparison of Interpolation Widths, Entropy Numbers, and Kolmogorov Widths

We compare the Kolmogorov and entropy numbers of compact operators mapping from a Hilbert space into a Banach space. We then apply these general findings to embeddings between reproducing kernel Hilbert spaces and $L_\infty(μ)$. Here we provide a sufficient condition for a gap of the order $n^{1/2}$ between the associated interpolation and Kolmogorov $n$-widths. Finally, we show that in the multi-dimensional Sobolev case, this gap actually occurs between the Kolmogorov and approximation widths.

math.FA↗

arXiv2016

Learning theory estimates with observations from general stationary stochastic processes

This paper investigates the supervised learning problem with observations drawn from certain general stationary stochastic processes. Here by \emph{general}, we mean that many stationary stochastic processes can be included. We show that when the stochastic processes satisfy a generalized Bernstein-type inequality, a unified treatment on analyzing the learning schemes with various mixing processes can be conducted and a sharp oracle inequality for generic regularized empirical risk minimization schemes can be established. The obtained oracle inequality is then applied to derive convergence rates for several learning schemes such as empirical risk minimization (ERM), least squares support vector machines (LS-SVMs) using given generic kernels, and SVMs using Gaussian kernels for both least squares and quantile regression. It turns out that for i.i.d.~processes, our learning rates for ERM recover the optimal rates. On the other hand, for non-i.i.d.~processes including geometrically $α$-mixing Markov processes, geometrically $α$-mixing processes with restricted decay, $ϕ$-mixing processes, and (time-reversed) geometrically $\mathcal{C}$-mixing processes, our learning rates for SVMs with Gaussian kernels match, up to some arbitrarily small extra term in the exponent, the optimal rates. For the remaining cases, our rates are at least close to the optimal rates. As a by-product, the assumed generalized Bernstein-type inequality also provides an interpretation of the so-called "effective number of observations" for various mixing processes.

stat.ML↗

arXiv2015

Fully adaptive density-based clustering

The clusters of a distribution are often defined by the connected components of a density level set. However, this definition depends on the user-specified level. We address this issue by proposing a simple, generic algorithm, which uses an almost arbitrary level set estimator to estimate the smallest level at which there are more than one connected components. In the case where this algorithm is fed with histogram-based level set estimates, we provide a finite sample analysis, which is then used to show that the algorithm consistently estimates both the smallest level and the corresponding connected components. We further establish rates of convergence for the two estimation problems, and last but not least, we present a simple, yet adaptive strategy for determining the width-parameter of the involved density estimator in a data-depending way.

stat.ME↗

arXiv2015

Representation of Quasi-Monotone Functionals by Families of Separating Hyperplanes

We characterize when the level sets of a continuous quasi-monotone functional defined on a suitable convex subset of a normed space can be uniquely represented by a family of bounded continuous functionals. Furthermore, we investigate how regularly these functionals depend on the parameterizing level. Finally, we show how this question relates to the recent problem of property elicitation that simultaneously attracted interest in machine learning, statistical evaluation of forecasts, and finance.

math.OC↗

arXiv2015

Towards an Axiomatic Approach to Hierarchical Clustering of Measures

We propose some axioms for hierarchical clustering of probability measures and investigate their ramifications. The basic idea is to let the user stipulate the clusters for some elementary measures. This is done without the need of any notion of metric, similarity or dissimilarity. Our main results then show that for each suitable choice of user-defined clustering on elementary measures we obtain a unique notion of clustering on a large set of distributions satisfying a set of additivity and continuity axioms. We illustrate the developed theory by numerous examples including some with and some without a density.

stat.ML↗

arXiv2015

Optimal Learning Rates for Localized SVMs

One of the limiting factors of using support vector machines (SVMs) in large scale applications are their super-linear computational requirements in terms of the number of training samples. To address this issue, several approaches that train SVMs on many small chunks of large data sets separately have been proposed in the literature. So far, however, almost all these approaches have only been empirically investigated. In addition, their motivation was always based on computational requirements. In this work, we consider a localized SVM approach based upon a partition of the input space. For this local SVM, we derive a general oracle inequality. Then we apply this oracle inequality to least squares regression using Gaussian kernels and deduce local learning rates that are essentially minimax optimal under some standard smoothness assumptions on the regression function. This gives the first motivation for using local SVMs that is not based on computational requirements but on theoretical predictions on the generalization performance. We further introduce a data-dependent parameter selection method for our local SVM approach and show that this method achieves the same learning rates as before. Finally, we present some larger scale experiments for our localized SVM showing that it achieves essentially the same test performance as a global SVM for a fraction of the computational requirements. In addition, it turns out that the computational requirements for the local SVMs are similar to those of a vanilla random chunk approach, while the achieved test errors are significantly better.

stat.ML↗

arXiv2015

An SVM-like Approach for Expectile Regression

Expectile regression is a nice tool for investigating conditional distributions beyond the conditional mean. It is well-known that expectiles can be described with the help of the asymmetric least square loss function, and this link makes it possible to estimate expectiles in a non-parametric framework by a support vector machine like approach. In this work we develop an efficient sequential-minimal-optimization-based solver for the underlying optimization problem. The behavior of the solver is investigated by conducting various experiments and the results are compared with the recent R-package ER-Boost.

stat.CO↗

arXiv2013

Hypothesis Testing for Validation and Certification

We develop a hypothesis testing framework for the formulation of the problems of 1) the validation of a simulation model and 2) using modeling to certify the performance of a physical system. These results are used to solve the extrapolative validation and certification problems, namely problems where the regime of interest is different than the regime for which we have experimental data. We use concentration of measure theory to develop the tests and analyze their errors. This work was stimulated by the work of Lucas, Owhadi, and Ortiz where a rigorous method of validation and certification is described and tested. In a remark we describe the connection between the two approaches. Moreover, as mentioned in that work these results have important implications in the Quantification of Margins and Uncertainties (QMU) framework. In particular, in a remark we describe how it provides a rigorous interpretation of the notion of confidence and new notions of margins and uncertainties which allow this interpretation. Since certain concentration parameters used in the above tests may be unkown, we furthermore show, in the last half of the paper, how to derive equally powerful tests which estimate them from sample data, thus replacing the assumption of the values of the concentration parameters with weaker assumptions. This paper is an essentially exact copy of one dated April 10, 2010.

stat.ME↗

arXiv2012

Forecasting with Historical Data or Process Knowledge under Misspecification: A Comparison

When faced with the task of forecasting a dynamic system, practitioners often have available historical data, knowledge of the system, or a combination of both. While intuition dictates that perfect knowledge of the system should in theory yield perfect forecasting, often knowledge of the system is only partially known, known up to parameters, or known incorrectly. In contrast, forecasting using previous data without any process knowledge might result in accurate prediction for simple systems, but will fail for highly nonlinear and chaotic systems. In this paper, the authors demonstrate how even in chaotic systems, forecasting with historical data is preferable to using process knowledge if this knowledge exhibits certain forms of misspecification. Through an extensive simulation study, a range of misspecification and forecasting scenarios are examined with the goal of gaining an improved understanding of the circumstances under which forecasting from historical data is to be preferred over using process knowledge.

stat.ME↗

arXiv2011

Estimating conditional quantiles with the help of the pinball loss

The so-called pinball loss for estimating conditional quantiles is a well-known tool in both statistics and machine learning. So far, however, only little work has been done to quantify the efficiency of this tool for nonparametric approaches. We fill this gap by establishing inequalities that describe how close approximate pinball risk minimizers are to the corresponding conditional quantile. These inequalities, which hold under mild assumptions on the data-generating distribution, are then used to establish so-called variance bounds, which recently turned out to play an important role in the statistical analysis of (regularized) empirical risk minimization approaches. Finally, we use both types of inequalities to establish an oracle inequality for support vector machines that use the pinball loss. The resulting learning rates are min--max optimal under some standard regularity assumptions on the conditional quantile.

math.ST↗

arXiv2009

Consistency of support vector machines for forecasting the evolution of an unknown ergodic dynamical system from observations with unknown noise

We consider the problem of forecasting the next (observable) state of an unknown ergodic dynamical system from a noisy observation of the present state. Our main result shows, for example, that support vector machines (SVMs) using Gaussian RBF kernels can learn the best forecaster from a sequence of noisy observations if (a) the unknown observational noise process is bounded and has a summable $α$-mixing rate and (b) the unknown ergodic dynamical system is defined by a Lipschitz continuous function on some compact subset of $\mathbb{R}^d$ and has a summable decay of correlations for Lipschitz continuous functions. In order to prove this result we first establish a general consistency result for SVMs and all stochastic processes that satisfy a mixing notion that is substantially weaker than $α$-mixing.

stat.ME↗

arXiv2007

Consistency and robustness of kernel-based regression in convex risk minimization

We investigate statistical properties for a broad class of modern kernel-based regression (KBR) methods. These kernel methods were developed during the last decade and are inspired by convex risk minimization in infinite-dimensional Hilbert spaces. One leading example is support vector regression. We first describe the relationship between the loss function $L$ of the KBR method and the tail of the response variable. We then establish the $L$-risk consistency for KBR which gives the mathematical justification for the statement that these methods are able to ``learn''. Then we consider robustness properties of such kernel methods. In particular, our results allow us to choose the loss function and the kernel to obtain computationally tractable and consistent KBR methods that have bounded influence functions. Furthermore, bounds for the bias and for the sensitivity curve, which is a finite sample version of the influence function, are developed, and the relationship between KBR and classical $M$ estimators is discussed.

math.ST↗

arXiv2007

Fast rates for support vector machines using Gaussian kernels

For binary classification we establish learning rates up to the order of $n^{-1}$ for support vector machines (SVMs) with hinge loss and Gaussian RBF kernels. These rates are in terms of two assumptions on the considered distributions: Tsybakov's noise assumption to establish a small estimation error, and a new geometric noise condition which is used to bound the approximation error. Unlike previously proposed concepts for bounding the approximation error, the geometric noise assumption does not employ any smoothness assumption.

math.ST↗

arXiv2007

Forecasting the Evolution of Dynamical Systems from Noisy Observations

We consider the problem of designing almost optimal predictors for dynamical systems from a finite sequence of noisy observations and incomplete knowledge of the dynamics and the noise. We first discuss the properties of the optimal (Bayes) predictor and present the limitations of memory-free forecasting methods, and of any finite memory methods in general. We then show that a nonparametric support vector machine approach to forecasting can consistently learn the optimal predictor for all pairs of dynamical systems and bounded observational noise processes that possess summable correlation sequences. Numerical experiments show that this approach adapts the memory length of the forecaster to the complexity of the learning task and the size of the observation sequence.

nlin.CD↗

arXiv2007

Learning from dependent observations

In most papers establishing consistency for learning algorithms it is assumed that the observations used for training are realizations of an i.i.d. process. In this paper we go far beyond this classical framework by showing that support vector machines (SVMs) essentially only require that the data-generating process satisfies a certain law of large numbers. We then consider the learnability of SVMs for $\a$-mixing (not necessarily stationary) processes for both classification and regression, where for the latter we explicitly allow unbounded noise.

stat.ML↗

arXiv2006

A new concentration result for regularized risk minimizers

We establish a new concentration result for regularized risk minimizers which is similar to an oracle inequality. Applying this inequality to regularized least squares minimizers like least squares support vector machines, we show that these algorithms learn with (almost) the optimal rate in some specific situations. In addition, for regression our results suggest that using the loss function $L_α(y,t)=|y-t|^α$ with $α$ near 1 may often be preferable to the usual choice of $α=2$.

math.ST↗