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Andreas Christmann

Andreas Christmann appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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Machine Learningmath.STStatistics Theorymath.FAMethodology

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Published work

11 published item(s)

preprint2026arXiv

Ratio-based Loss Functions

Algorithms in machine learning and AI do critically depend on at least three key components: (i) the risk function, which is the expectation of the loss function, (ii) the function space, which is often called the hypothesis space, and (iii) the set of probability measures, which are allowed for the specified algorithm. This paper gives a survey of a certain class of loss functions, which we call ratio-based. In supervised learning, margin-based loss functions for classification tasks depending on the product of the output values $y_i$ and the predictions $f(x_i)$ as well as distance-based loss functions depending on the difference of $y_i$ and $f(x_i)$ for regression are common. Distance-based loss functions are in particular useful, if an additive model assumption seems plausible, i.e. the common signal plus noise assumption. However, in the literature, several loss functions proposed for regression purposes have a multiplicative error structure in mind and pay attention to relative errors, i.e. to the ratio of $y_i$ and $f(x_i)$. In this survey article, we systematically investigate such ratio-based loss functions and propose a few new losses, which may be interesting for future research. We concentrate on investigating general properties of ratio-based loss functions like continuity, Lipschitz-continuity, convexity, and differentiability, because these properties play a central role in most machine learning algorithms. Therefore, we do not focus on some specific machine learning algorithm to derive universal consistency, learning rates, or stability results. Instead, we want to enable future research in this direction.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Total Stability of SVMs and Localized SVMs

Regularized kernel-based methods such as support vector machines (SVMs) typically depend on the underlying probability measure $\mathrm{P}$ (respectively an empirical measure $\mathrm{D}_n$ in applications) as well as on the regularization parameter $λ$ and the kernel $k$. Whereas classical statistical robustness only considers the effect of small perturbations in $\mathrm{P}$, the present paper investigates the influence of simultaneous slight variations in the whole triple $(\mathrm{P},λ,k)$, respectively $(\mathrm{D}_n,λ_n,k)$, on the resulting predictor. Existing results from the literature are considerably generalized and improved. In order to also make them applicable to big data, where regular SVMs suffer from their super-linear computational requirements, we show how our results can be transferred to the context of localized learning. Here, the effect of slight variations in the applied regionalization, which might for example stem from changes in $\mathrm{P}$ respectively $\mathrm{D}_n$, is considered as well.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

A short note on extension theorems and their connection to universal consistency in machine learning

Statistical machine learning plays an important role in modern statistics and computer science. One main goal of statistical machine learning is to provide universally consistent algorithms, i.e., the estimator converges in probability or in some stronger sense to the Bayes risk or to the Bayes decision function. Kernel methods based on minimizing the regularized risk over a reproducing kernel Hilbert space (RKHS) belong to these statistical machine learning methods. It is in general unknown which kernel yields optimal results for a particular data set or for the unknown probability measure. Hence various kernel learning methods were proposed to choose the kernel and therefore also its RKHS in a data adaptive manner. Nevertheless, many practitioners often use the classical Gaussian RBF kernel or certain Sobolev kernels with good success. The goal of this short note is to offer one possible theoretical explanation for this empirical fact.

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preprint2015arXiv

On the Robustness of Regularized Pairwise Learning Methods Based on Kernels

Regularized empirical risk minimization including support vector machines plays an important role in machine learning theory. In this paper regularized pairwise learning (RPL) methods based on kernels will be investigated. One example is regularized minimization of the error entropy loss which has recently attracted quite some interest from the viewpoint of consistency and learning rates. This paper shows that such RPL methods have additionally good statistical robustness properties, if the loss function and the kernel are chosen appropriately. We treat two cases of particular interest: (i) a bounded and non-convex loss function and (ii) an unbounded convex loss function satisfying a certain Lipschitz type condition.

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preprint2014arXiv

Learning rates for the risk of kernel based quantile regression estimators in additive models

Additive models play an important role in semiparametric statistics. This paper gives learning rates for regularized kernel based methods for additive models. These learning rates compare favourably in particular in high dimensions to recent results on optimal learning rates for purely nonparametric regularized kernel based quantile regression using the Gaussian radial basis function kernel, provided the assumption of an additive model is valid. Additionally, a concrete example is presented to show that a Gaussian function depending only on one variable lies in a reproducing kernel Hilbert space generated by an additive Gaussian kernel, but does not belong to the reproducing kernel Hilbert space generated by the multivariate Gaussian kernel of the same variance.

0 citations0 reviewsNo code linkedOpen access
preprint2011arXiv

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.

0 citations0 reviewsNo code linkedOpen access
preprint2011arXiv

Estimation of scale functions to model heteroscedasticity by support vector machines

A main goal of regression is to derive statistical conclusions on the conditional distribution of the output variable Y given the input values x. Two of the most important characteristics of a single distribution are location and scale. Support vector machines (SVMs) are well established to estimate location functions like the conditional median or the conditional mean. We investigate the estimation of scale functions by SVMs when the conditional median is unknown, too. Estimation of scale functions is important e.g. to estimate the volatility in finance. We consider the median absolute deviation (MAD) and the interquantile range (IQR) as measures of scale. Our main result shows the consistency of MAD-type SVMs.

0 citations0 reviewsNo code linkedOpen access
preprint2011arXiv

Qualitative Robustness of Support Vector Machines

Support vector machines have attracted much attention in theoretical and in applied statistics. Main topics of recent interest are consistency, learning rates and robustness. In this article, it is shown that support vector machines are qualitatively robust. Since support vector machines can be represented by a functional on the set of all probability measures, qualitative robustness is proven by showing that this functional is continuous with respect to the topology generated by weak convergence of probability measures. Combined with the existence and uniqueness of support vector machines, our results show that support vector machines are the solutions of a well-posed mathematical problem in Hadamard's sense.

0 citations0 reviewsNo code linkedOpen access
preprint2010arXiv

Support Vector Machines for Additive Models: Consistency and Robustness

Support vector machines (SVMs) are special kernel based methods and belong to the most successful learning methods since more than a decade. SVMs can informally be described as a kind of regularized M-estimators for functions and have demonstrated their usefulness in many complicated real-life problems. During the last years a great part of the statistical research on SVMs has concentrated on the question how to design SVMs such that they are universally consistent and statistically robust for nonparametric classification or nonparametric regression purposes. In many applications, some qualitative prior knowledge of the distribution P or of the unknown function f to be estimated is present or the prediction function with a good interpretability is desired, such that a semiparametric model or an additive model is of interest. In this paper we mainly address the question how to design SVMs by choosing the reproducing kernel Hilbert space (RKHS) or its corresponding kernel to obtain consistent and statistically robust estimators in additive models. We give an explicit construction of kernels - and thus of their RKHSs - which leads in combination with a Lipschitz continuous loss function to consistent and statistically robust SMVs for additive models. Examples are quantile regression based on the pinball loss function, regression based on the epsilon-insensitive loss function, and classification based on the hinge loss function.

0 citations0 reviewsNo code linkedOpen access