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Reinforcement learning (RL) algorithms based on high-dimensional function approximation have achieved tremendous empirical success in large-scale problems with an enormous number of states. However, most analysis of such algorithms gives…

We study generalization properties of distributed algorithms in the setting of nonparametric regression over a reproducing kernel Hilbert space (RKHS). We first investigate distributed stochastic gradient methods (SGM), with mini-batches…

We introduce a semi-supervised learning estimator which tends to the first kernel principal component as the number of labelled points vanishes. Our approach is based on the notion of optimal target vector, which is defined as follows.…

Linear fixed point equations in Hilbert spaces arise in a variety of settings, including reinforcement learning, and computational methods for solving differential and integral equations. We study methods that use a collection of random…

Despite the ubiquity of the Gaussian process regression model, few theoretical results are available that account for the fact that parameters of the covariance kernel typically need to be estimated from the dataset. This article provides…

Let $\Omega\subset \mathbb{R}^d$ be a bounded domain. We consider the problem of how efficiently shallow neural networks with the ReLU$^k$ activation function can approximate functions from Sobolev spaces $W^s(L_p(\Omega))$ with error…

We develop a comprehensive framework for spatio-temporal prediction of time-varying vector fields using operator-valued reproducing kernel Hilbert spaces (OV RKHS). By integrating Sobolev regularity with Koopman operator theory, we…

The Ridgeless minimum $\ell_2$-norm interpolator in overparametrized linear regression has attracted considerable attention in recent years in both machine learning and statistics communities. While it seems to defy conventional wisdom that…

This paper studies convergence rates for some value function approximations that arise in a collection of reproducing kernel Hilbert spaces (RKHS) $H(\Omega)$. By casting an optimal control problem in a specific class of native spaces,…

In this paper, we investigate the supremum-norm generalization error and the uniform inference for a specific class of kernel regression methods, namely the kernel gradient flows. Under the widely adopted capacity-source condition framework…

A typical approach in estimating the learning rate of a regularized learning scheme is to bound the approximation error by the sum of the sampling error, the hypothesis error and the regularization error. Using a reproducing kernel space…

The support vector machine (SVM) is a supervised learning algorithm that finds a maximum-margin linear classifier, often after mapping the data to a high-dimensional feature space via the kernel trick. Recent work has demonstrated that in…

Classical convergence analysis for kernel interpolation typically assumes that the target function $f$ lies in the reproducing kernel Hilbert space $\mathcal{H}_k\!\left(\Omega\right)$ induced by a kernel on a domain…

We develop nonparametric regression methods for the case when the true regression function is not necessarily smooth. More specifically, our approach is using the fractional Laplacian and is designed to handle the case when the true…

Let $\Lambda \subset R$ be a strictly increasing sequence. For $r = 1,2$, we give a simple explicit expression for an equivalent norm on the trace spaces $W_p^r(R)|_\Lambda$, $L_p^r(R)|_\Lambda$ of the non-homogeneous and homogeneous…

We study Tikhonov regularization for possibly nonlinear inverse problems with weighted $\ell^1$-penalization. The forward operator, mapping from a sequence space to an arbitrary Banach space, typically an $L^2$-space, is assumed to satisfy…

We provide uniform confidence bands for kernel ridge regression (KRR), a widely used nonparametric regression estimator for nonstandard data such as preferences, sequences, and graphs. Despite the prevalence of these data--e.g., student…

The problem of establishing out-of-sample bounds for the values of an unkonwn ground-truth function is considered. Kernels and their associated Hilbert spaces are the main formalism employed herein along with an observational model where…

We study the implicit regularization of optimization methods for linear models interpolating the training data in the under-parametrized and over-parametrized regimes. Since it is difficult to determine whether an optimizer converges to…

We study nonlinear regression of real valued data in an individual sequence manner, where we provide results that are guaranteed to hold without any statistical assumptions. We address the convergence and undertraining issues of…