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Ingo Steinwart

Publications and source records attributed to Ingo Steinwart.

At least 37 records · Page 2Linked to original sources
arXiv2021

Empirical Risk Minimization in the Interpolating Regime with Application to Neural Network Learning

A common strategy to train deep neural networks (DNNs) is to use very large architectures and to train them until they (almost) achieve zero training error. Empirically observed good generalization performance on test data, even in the presence of lots of label noise, corroborate such a procedure. On the other hand, in statistical learning theory it is known that over-fitting models may lead to poor generalization properties, occurring in e.g. empirical risk minimization (ERM) over too large hypotheses classes. Inspired by this contradictory behavior, so-called interpolation methods have recently received much attention, leading to consistent and optimally learning methods for some local averaging schemes with zero training error. However, there is no theoretical analysis of interpolating ERM-like methods so far. We take a step in this direction by showing that for certain, large hypotheses classes, some interpolating ERMs enjoy very good statistical guarantees while others fail in the worst sense. Moreover, we show that the same phenomenon occurs for DNNs with zero training error and sufficiently large architectures.

stat.ML↗
arXiv2021

Intrinsic Dimension Adaptive Partitioning for Kernel Methods

We prove minimax optimal learning rates for kernel ridge regression, resp.~support vector machines based on a data dependent partition of the input space, where the dependence of the dimension of the input space is replaced by the fractal dimension of the support of the data generating distribution. We further show that these optimal rates can be achieved by a training validation procedure without any prior knowledge on this intrinsic dimension of the data. Finally, we conduct extensive experiments which demonstrate that our considered learning methods are actually able to generalize from a dataset that is non-trivially embedded in a much higher dimensional space just as well as from the original dataset.

math.ST↗
arXiv2021

Adaptive Learning Rates for Support Vector Machines Working on Data with Low Intrinsic Dimension

We derive improved regression and classification rates for support vector machines using Gaussian kernels under the assumption that the data has some low-dimensional intrinsic structure that is described by the box-counting dimension. Under some standard regularity assumptions for regression and classification we prove learning rates, in which the dimension of the ambient space is replaced by the box-counting dimension of the support of the data generating distribution. In the regression case our rates are in some cases minimax optimal up to logarithmic factors, whereas in the classification case our rates are minimax optimal up to logarithmic factors in a certain range of our assumptions and otherwise of the form of the best known rates. Furthermore, we show that a training validation approach for choosing the hyperparameters of an SVM in a data dependent way achieves the same rates adaptively, that is without any knowledge on the data generating distribution.

math.ST↗
arXiv2020

Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm

Learning rates for least-squares regression are typically expressed in terms of $L_2$-norms. In this paper we extend these rates to norms stronger than the $L_2$-norm without requiring the regression function to be contained in the hypothesis space. In the special case of Sobolev reproducing kernel Hilbert spaces used as hypotheses spaces, these stronger norms coincide with fractional Sobolev norms between the used Sobolev space and $L_2$. As a consequence, not only the target function but also some of its derivatives can be estimated without changing the algorithm. From a technical point of view, we combine the well-known integral operator techniques with an embedding property, which so far has only been used in combination with empirical process arguments. This combination results in new finite sample bounds with respect to the stronger norms. From these finite sample bounds our rates easily follow. Finally, we prove the asymptotic optimality of our results in many cases.

stat.ML↗
arXiv2020

A Closer Look at Covering Number Bounds for Gaussian Kernels

We establish some new bounds on the log-covering numbers of (anisotropic) Gaussian reproducing kernel Hilbert spaces. Unlike previous results in this direction we focus on small explicit constants and their dependency on crucial parameters such as the kernel bandwidth and the size and dimension of the underlying space.

math.FA↗
arXiv2019

Improved Classification Rates for Localized SVMs

Localized support vector machines solve SVMs on many spatially defined small chunks and one of their main characteristics besides the computational benefit compared to global SVMs is the freedom of choosing arbitrary kernel and regularization parameter on each cell. We take advantage of this observation to derive global learning rates for localized SVMs with Gaussian kernels and hinge loss. Under certain assumptions our rates outperform known classification rates for localized SVMs, for global SVMs, and other learning algorithms based on e.g., plug-in rules, trees, or DNNs. These rates are achieved under a set of margin conditions that describe the behavior of the data-generating distribution, where no assumption on the existence of a density is made. We observe that a margin condition that relates the distance to the decision boundary to the amount of noise is crucial to obtain rates. The statistical analysis relies on a careful analysis of the excess risk which includes a separation of the input space into a subset that is close to the decision boundary and into a subset that is sufficiently far away. Moreover, we show that our rates are obtained adaptively, that is, without knowing the parameters resulting from the margin conditions.

math.ST↗
arXiv2019

PAC-Bayesian Bounds for Deep Gaussian Processes

Variational approximation techniques and inference for stochastic models in machine learning has gained much attention the last years. Especially in the case of Gaussian Processes (GP) and their deep versions, Deep Gaussian Processes (DGPs), these viewpoints improved state of the art work. In this paper we introduce Probably Approximately Correct (PAC)-Bayesian risk bounds for DGPs making use of variational approximations. We show that the minimization of PAC-Bayesian generalization risk bounds maximizes the variational lower bounds belonging to the specific DGP model. We generalize the loss function property of the log likelihood loss function in the context of PAC-Bayesian risk bounds to the quadratic-form-Gaussian case. Consistency results are given and an oracle-type inequality gives insights in the convergence between the raw model (predictor without variational approximation) and our variational models (predictor for the variational approximation). Furthermore, we give extensions of our main theorems for specific assumptions and parameter cases. Moreover, we show experimentally the evolution of the consistency results for two Deep Recurrent Gaussian Processes (DRGP) modeling time-series, namely the recurrent Gaussian Process (RGP) and the DRGP with Variational Sparse Spectrum approximation, namely DRGP-(V)SS.

math.ST↗
arXiv2019

A Sober Look at Neural Network Initializations

Initializing the weights and the biases is a key part of the training process of a neural network. Unlike the subsequent optimization phase, however, the initialization phase has gained only limited attention in the literature. In this paper we discuss some consequences of commonly used initialization strategies for vanilla DNNs with ReLU activations. Based on these insights we then develop an alternative initialization strategy. Finally, we present some large scale experiments assessing the quality of the new initialization strategy.

cs.LG↗
arXiv2019

Best-scored Random Forest Classification

We propose an algorithm named best-scored random forest for binary classification problems. The terminology "best-scored" means to select the one with the best empirical performance out of a certain number of purely random tree candidates as each single tree in the forest. In this way, the resulting forest can be more accurate than the original purely random forest. From the theoretical perspective, within the framework of regularized empirical risk minimization penalized on the number of splits, we establish almost optimal convergence rates for the proposed best-scored random trees under certain conditions which can be extended to the best-scored random forest. In addition, we present a counterexample to illustrate that in order to ensure the consistency of the forest, every dimension must have the chance to be split. In the numerical experiments, for the sake of efficiency, we employ an adaptive random splitting criterion. Comparative experiments with other state-of-art classification methods demonstrate the accuracy of our best-scored random forest.

stat.ML↗
arXiv2018

Optimal Learning with Anisotropic Gaussian SVMs

This paper investigates the nonparametric regression problem using SVMs with anisotropic Gaussian RBF kernels. Under the assumption that the target functions are resided in certain anisotropic Besov spaces, we establish the almost optimal learning rates, more precisely, optimal up to some logarithmic factor, presented by the effective smoothness. By taking the effective smoothness into consideration, our almost optimal learning rates are faster than those obtained with the underlying RKHSs being certain anisotropic Sobolev spaces. Moreover, if the target function depends only on fewer dimensions, faster learning rates can be further achieved.

stat.ML↗
arXiv2018

Spatial Decompositions for Large Scale SVMs

Although support vector machines (SVMs) are theoretically well understood, their underlying optimization problem becomes very expensive, if, for example, hundreds of thousands of samples and a non-linear kernel are considered. Several approaches have been proposed in the past to address this serious limitation. In this work we investigate a decomposition strategy that learns on small, spatially defined data chunks. Our contributions are two fold: On the theoretical side we establish an oracle inequality for the overall learning method using the hinge loss, and show that the resulting rates match those known for SVMs solving the complete optimization problem with Gaussian kernels. On the practical side we compare our approach to learning SVMs on small, randomly chosen chunks. Here it turns out that for comparable training times our approach is significantly faster during testing and also reduces the test error in most cases significantly. Furthermore, we show that our approach easily scales up to 10 million training samples: including hyper-parameter selection using cross validation, the entire training only takes a few hours on a single machine. Finally, we report an experiment on 32 million training samples. All experiments used liquidSVM (Steinwart and Thomann, 2017).

stat.ML↗
arXiv2018

Improved Classification Rates under Refined Margin Conditions

In this paper we present a simple partitioning based technique to refine the statistical analysis of classification algorithms. The core of our idea is to divide the input space into two parts such that the first part contains a suitable vicinity around the decision boundary, while the second part is sufficiently far away from the decision boundary. Using a set of margin conditions we are then able to control the classification error on both parts separately. By balancing out these two error terms we obtain a refined error analysis in a final step. We apply this general idea to the histogram rule and show that even for this simple method we obtain, under certain assumptions, better rates than the ones known for support vector machines, for certain plug-in classifiers, and for a recently analyzed tree based adaptive-partitioning ansatz. Moreover, we show that a margin condition which sets the critical noise in relation to the decision boundary makes it possible to improve the optimal rates proven for distributions without this margin condition.

math.ST↗
arXiv2017

Strictly proper kernel scores and characteristic kernels on compact spaces

Strictly proper kernel scores are well-known tool in probabilistic forecasting, while characteristic kernels have been extensively investigated in the machine learning literature. We first show that both notions coincide, so that insights from one part of the literature can be used in the other. We then show that the metric induced by a characteristic kernel cannot reliably distinguish between distributions that are far apart in the total variation norm as soon as the underlying space of measures is infinite dimensional. In addition, we provide a characterization of characteristic kernels in terms of eigenvalues and -functions and apply this characterization to the case of continuous kernels on (locally) compact spaces. In the compact case we further show that characteristic kernels exist if and only if the space is metrizable. As special cases of our general theory we investigate translation-invariant kernels on compact Abelian groups and isotropic kernels on spheres. The latter are of particular interest for forecast evaluation of probabilistic predictions on spherical domains as frequently encountered in meteorology and climatology.

math.FA↗
arXiv2017

Convergence Types and Rates in Generic Karhunen-Loève Expansions with Applications to Sample Path Properties

We establish a Karhunen-Lo`eve expansion for generic centered, second order stochastic processes, which does not rely on topological assumptions. We further investigate in which norms the expansion converges and derive exact average rates of convergence for these norms. For Gaussian processes we additionally prove certain sharpness results in terms of the norm. Moreover, we show that the generic Karhunen-Lo`eve expansion can in some situations be used to construct reproducing kernel Hilbert spaces (RKHSs) containing the paths of a version of the process. As applications of the general theory, we compare the smoothness of the paths with the smoothness of the functions contained in the RKHS of the covariance function and discuss some small ball probabilities. Key tools for our results are a recently shown generalization of Mercer's theorem, spectral properties of the covariance integral operator, interpolation spaces of the real method, and for the smoothness results, entropy numbers of embeddings between classical function spaces.

math.PR↗
arXiv2017

Learning Rates for Kernel-Based Expectile Regression

Conditional expectiles are becoming an increasingly important tool in finance as well as in other areas of applications. We analyse a support vector machine type approach for estimating conditional expectiles and establish learning rates that are minimax optimal modulo a logarithmic factor if Gaussian RBF kernels are used and the desired expectile is smooth in a Besov sense. As a special case, our learning rates improve the best known rates for kernel-based least squares regression in this scenario. Key ingredients of our statistical analysis are a general calibration inequality for the asymmetric least squares loss, a corresponding variance bound as well as an improved entropy number bound for Gaussian RBF kernels.

stat.ML↗
arXiv2017

liquidSVM: A Fast and Versatile SVM package

liquidSVM is a package written in C++ that provides SVM-type solvers for various classification and regression tasks. Because of a fully integrated hyper-parameter selection, very carefully implemented solvers, multi-threading and GPU support, and several built-in data decomposition strategies it provides unprecedented speed for small training sizes as well as for data sets of tens of millions of samples. Besides the C++ API and a command line interface, bindings to R, MATLAB, Java, Python, and Spark are available. We present a brief description of the package and report experimental comparisons to other SVM packages.

stat.ML↗
arXiv2016

Learning with Hierarchical Gaussian Kernels

We investigate iterated compositions of weighted sums of Gaussian kernels and provide an interpretation of the construction that shows some similarities with the architectures of deep neural networks. On the theoretical side, we show that these kernels are universal and that SVMs using these kernels are universally consistent. We further describe a parameter optimization method for the kernel parameters and empirically compare this method to SVMs, random forests, a multiple kernel learning approach, and to some deep neural networks.

stat.ML↗