Marcin Wnuk

Michael Gnewuch, Marcin Wnuk

Smolyak's method, also known as hyperbolic cross approximation or sparse grid method, is a powerful tool to tackle multivariate tensor product problems solely with the help of efficient algorithms for the corresponding univariate problem. In this paper we study the randomized setting, i.e., we randomize Smolyak's method. We provide upper and lower error bounds for randomized Smolyak algorithms with explicitly given dependence on the number of variables and the number of information evaluations used. The error criteria we consider are the worst-case root mean square error (the typical error criterion for randomized algorithms, often referred to as "randomized error") and the root mean square worst-case error (often referred to as "worst-case error"). Randomized Smolyak algorithms can be used as building blocks for efficient methods such as multilevel algorithms, multivariate decomposition methods or dimension-wise quadrature methods to tackle successfully high-dimensional or even infnite-dimensional problems. As an example, we provide a very general and sharp result on the convergence rate of N-th minimal errors of infnite-dimensional integration on weighted reproducing kernel Hilbert spaces. Moreover, we are able to characterize the spaces for which randomized algorithms for infnte-dimensional integration are superior to deterministic ones. We illustrate our fndings for the special instance of weighted Korobov spaces. We indicate how these results can be extended, e.g., to spaces of functions whose smooth dependence on successive variables increases ("spaces of increasing smoothness") and to the problem of L2-approximation (function recovery).

Marcin Wnuk, Michael Gnewuch

In her recent paper [Negative dependence, scrambled nets, and variance bounds. Math. Oper. Res. 43 (2018), 228-251] Christiane Lemieux studied a framework to analyze the dependence structure of sampling schemes. The main goal of the framework is to determine conditions under which the negative dependence structure of a sampling scheme yields estimators with reduced variance compared to Monte Carlo estimators. For instance, she was able to show that in dimension d = 2 scrambled (0,m, d)-nets lead to randomized quasi-Monte Carlo estimators with variance no larger than the variance of Monte Carlo estimators for functions monotone in each variable. Her result relies on a pairwise negative dependence property that is, in particular, satisfied by (0,m, 2)-nets. In this note we establish that the same result holds true in arbitrary dimension d for a type of randomized lattice point sets that we call randomly shifted and jittered rank-1 lattices. We show that the details of the randomization are crucial and that already small modifications may destroy the pairwise negative dependence property.

Michael Gnewuch, Kumar Harsha, Marcin Wnuk

The mean squared error and regularized versions of it are standard loss functions in supervised machine learning. However, calculating these losses for large data sets can be computationally demanding. Modifying an approach of J. Dick and M. Feischl [Journal of Complexity 67 (2021)], we present algorithms to reduce extensive data sets to a smaller size using rank-1 lattices. Rank-1 lattices are quasi-Monte Carlo (QMC) point sets that are, if carefully chosen, well-distributed in a multidimensional unit cube. The compression strategy in the preprocessing step assigns every lattice point a pair of weights depending on the original data and responses, representing its relative importance. As a result, the compressed data makes iterative loss calculations in optimization steps much faster. We analyze the errors of our QMC data compression algorithms and the cost of the preprocessing step for functions whose Fourier coefficients decay sufficiently fast so that they lie in certain Wiener algebras or Korobov spaces. In particular, we prove that our approach can lead to arbitrary high convergence rates as long as the functions are sufficiently smooth.