arXiv subjects

Paul Dommel

Publications and source records attributed to Paul Dommel.

6 recordsLinked to original sources

arXiv2025

Uniform convergence for Gaussian kernel ridge regression

This paper establishes the first polynomial convergence rates for Gaussian kernel ridge regression (KRR) with a fixed hyperparameter in both the uniform and the $L^{2}$-norm. The uniform convergence result closes a gap in the theoretical understanding of KRR with the Gaussian kernel, where no such rates were previously known. In addition, we prove a polynomial $L^{2}$-convergence rate in the case, where the Gaussian kernel's width parameter is fixed. This also contributes to the broader understanding of smooth kernels, for which previously only sub-polynomial $L^{2}$-rates were known in similar settings. Together, these results provide new theoretical justification for the use of Gaussian KRR with fixed hyperparameters in nonparametric regression.

stat.ML↗

arXiv2024

On the Approximation of Kernel functions

Various methods in statistical learning build on kernels considered in reproducing kernel Hilbert spaces. In applications, the kernel is often selected based on characteristics of the problem and the data. This kernel is then employed to infer response variables at points, where no explanatory data were observed. The data considered here are located in compact sets in higher dimensions and the paper addresses approximations of the kernel itself. The new approach considers Taylor series approximations of radial kernel functions. For the Gauss kernel on the unit cube, the paper establishes an upper bound of the associated eigenfunctions, which grows only polynomially with respect to the index. The novel approach substantiates smaller regularization parameters than considered in the literature, overall leading to better approximations. This improvement confirms low rank approximation methods such as the Nystr\"om method.

stat.ML↗

arXiv2024

A Bound on the Maximal Marginal Degrees of Freedom

Kernel ridge regression, in general, is expensive in memory allocation and computation time. This paper addresses low rank approximations and surrogates for kernel ridge regression, which bridge these difficulties. The fundamental contribution of the paper is a lower bound on the minimal rank such that the prediction power of the approximation remains reliable. Based on this bound, we demonstrate that the computational cost of the most popular low rank approach, which is the Nystr\"om method, is almost linear in the sample size. This justifies the method from a theoretical point of view. Moreover, the paper provides a significant extension of the feasible choices of the regularization parameter. The result builds on a thorough theoretical analysis of the approximation of elementary kernel functions by elements in the range of the associated integral operator. We provide estimates of the approximation error and characterize the behavior of the norm of the underlying weight function.

stat.ML↗

arXiv2021

Uniform Function Estimators in Reproducing Kernel Hilbert Spaces

This paper addresses the problem of regression to reconstruct functions, which are observed with superimposed errors at random locations. We address the problem in reproducing kernel Hilbert spaces. It is demonstrated that the estimator, which is often derived by employing Gaussian random fields, converges in the mean norm of the reproducing kernel Hilbert space to the conditional expectation and this implies local and uniform convergence of this function estimator. By preselecting the kernel, the problem does not suffer from the curse of dimensionality. The paper analyzes the statistical properties of the estimator. We derive convergence properties and provide a conservative rate of convergence for increasing sample sizes.

math.ST↗

arXiv2021

Foundations of Multistage Stochastic Programming

Multistage stochastic optimization problems are oftentimes formulated informally in a pathwise way. These are correct in a discrete setting and suitable when addressing computational challenges, for example. But the pathwise problem statement does not allow an analysis with mathematical rigor and is therefore not appropriate. This paper addresses the foundations. We provide a novel formulation of multistage stochastic optimization problems by involving adequate stochastic processes as control. The fundamental contribution is a proof that there exist measurable versions of intermediate value functions. Our proof builds on the Kolmogorov continuity theorem. A verification theorem is given in addition, and it is demonstrated that all traditional problem specifications can be stated in the novel setting with mathematical rigor. Further, we provide dynamic equations for the general problem, which is developed for various problem classes. The problem classes covered here include Markov decision processes, reinforcement learning and stochastic dual dynamic programming.

math.OC↗

arXiv2020

Convex Risk Measures based on Divergence

Risk measures connect probability theory or statistics to optimization, particularly to convex optimization. They are nowadays standard in applications of finance and in insurance involving risk aversion. This paper investigates a wide class of risk measures on Orlicz spaces. The characterizing function describes the decision maker's risk assessment towards increasing losses. We link the risk measures to a crucial formula developed by Rockafellar for the Average Value-at-Risk based on convex duality, which is fundamental in corresponding optimization problems. We characterize the dual and provide complementary representations.

q-fin.RM↗