# Research Group "Stochastic Algorithms and Nonparametric Statistics"

## Seminar "Modern Methods in Applied Stochastics and Nonparametric Statistics" Winter Semester 2018/2019

 Place: Weierstrass-Institute for Applied Analysis and Stochastics, Room 406 (4th floor), Mohrenstraße 39, 10117 Berlin Time: Tuesdays, 3:00PM - 4:00PM 11.09.2018 Franz Besold (Humboldt-Universität zu Berlin) Adaptive clustering using kernel density estimators We investigate statistical properties of a clustering algorithm suggested by Ingo Steinwart, Bharath K. Sriperumbudur and Philipp Thomann that receives level set estimates from a kernel density estimator and then estimates the first split in the density level cluster tree if such a split is present or detects the absence of such a split. Key aspects of the analysis include finite sample guarantees, consistency, rates of convergence, and an adaptive data-driven strategy for choosing the kernel bandwidth. For the rates and the adaptivity we do not need continuity assumptions on the density such as Hölder continuity, but only require intuitive geometric assumptions of non-parametric nature. 18.09.2018 Prof. Thamban Nair (Indian Institute of Technology Madras) An inverse problem in parabolic PDEs 25.09.2018 n. n. Seminar room no. 3.13 at HVP 11a 02.10.18 n. n. 09.10.2018 n. n. Seminar room no. 3.13 at HVP 11a 16.10.18 Alexander Gasnikov (MIPT) Unified view on accelerated methods for structural convex optimization problems 23.10.18 Alexey Naumov (Skoltech) Gaussian approximations for maxima of large number of quadratic forms of high-dimensional random vectors Let X_1, ... , X_n be i.i.d. random vectors taking values in R^d, d \geq 1, and Q_1, ... , Q_p, p \geq 1, be symmetric positive definite matrices. We consider the distribution function of vector (Q_j S_n, S_n), j = 1, ... , p, where S_n = n^{-1/2}(X_1 + ... + X_n), and prove the rate of Gaussian approximation with explicit dependence on n, p and d. We also compare this result with results of Bentkus (2003) and Chernozhukov, Chetverikov, Kato (2016). Applications to change point detection and model selection will be discussed as well. The talk is based on the joint project with F. Goetze, V. Spokoiny and A. Tikhomirov. 30.10.18 n. n. 06.11.18 John Meddocks (École Polytechnique Fédérale de Lausanne) First Talk 2 p.m. Estimating structured precision matrices In my group's work on estimating parameter sets in coarse-grain (or multi-scale) sequence-dependent models of DNA we came across the following basic problem: For a given (positive-definite) covariance matrix (in our case estimated in standard ways from Molecular dynamics simulation time series) what is the "best fit" precision, or inverse covariance, matrix under the contraint that the precision matrix must have a prescribed block banded sparsity pattern. I will describe various known, but apparently not well known, results in this and related directions, involving maximum entropy and relative entropy and likelihood principles. 06.11.18 Raul Tempone (RWTH Aachen) Second Talk 3 p.m. Multilevel and Multi-index Monte Carlo methods for the McKean-Vlasov equation In this talk, I will talk about our recent work where we address the approximation of functionals depending on a system of parti cles, described by stochastic differential equations (SDEs), in the mean-f ield limit when the number of particles is infinite. This problem is equiv alent to estimating the weak solution of the limiting McKean-Vlasov SDE. To that e nd, our approach uses systems with finite numbers of particles and a time-ste pping scheme. In this setting, there are two discretization parameters: the number of time steps and the number of particles. Based on these two parameters, we consider different variants of the Monte Carlo, Multilevel Monte Carlo (MLMC) and Multi-Index Monte Carlo methods and show that, based on som e assumptions that are verified numerically, we are able to achi eve a near- optimal work complexity in a typical setting. 13.11.18 Eric Joseph Hall (RWTH Aachen) Seminar room no. 3.13 at HVP 11a Causality and Bayesian network PDE for multiscale representations of porous media Hierarchical porous media that feature properties and processes at multiple scales arise in many engineering applications including the design of novel materials for energy storage devices. Microscopic (pore-scale) properties of the media impact their macroscopic (continuum- or Darcy-scale) counterparts and understanding the relationships between processes on these two scales is essential for informing engineering decision tasks. However, microscopic properties typically exhibit complex statistical correlations that present challenges for the estimation of macroscopic quantities of interest (QoIs), e.g., global sensitivity analysis (GSA) of macroscopic QoIs with respect to microscopic material properties that respect structural constraints. We present a systematic framework for building correlations into stochastic multiscale models through Bayesian networks. This allows us to construct the joint probability density function (PDF) of model parameters through causal relationships that emulate engineering processes related to the design of hierarchical nanoporous materials. Such PDFs also serve as input for the forward propagation of parametric uncertainty; our findings indicate that the inclusion of causal relationships impacts predictions of macroscopic QoIs. To assess the impact of correlations and causal relationships between microscopic parameters on macroscopic material properties, we use a moment-independent GSA based on the differential mutual information. Our GSA accounts for the correlated inputs and complex non-Gaussian QoIs. The global sensitivity indices are used to rank the effect of uncertainty in microscopic parameters on macroscopic QoIs, to quantify the impact of causality on the multiscale model's predictions, and to provide physical interpretations of these results for hierarchical nanoporous materials. 20.11.18 Michele Coghi (WIAS Berlin) Pathwise McKean-Vlasov theory We take a pathwise approach to classical McKean-Vlasov stochastic differential equations, as found e.g. in Sznitmann. We are much inspired by Cass-Lyons (and more recent works of Bailleul et al.), but avoid all rough path complications due to our focus on additive noise. The resulting ``pathwise McKean-Vlasov theory'' is both simple and powerful: as applications we discuss propagation of chaos with a priori independence and exchangeability assumption; common noise and reflecting boundaries are also easy to handle in this framework, last not least we can generalise the Dawson-Gärtner large deviation result to non-Brownian noise. 27.11.18 Yegor Klochkov (Humboldt Universität zu Berlin) On uniform Hanson-Wright type inequalities for sub-Gaussian entries We investigate concentration of supremum of quadratic forms via entropy inequality. The problem is well understood for Gaussian vectors, and, more generally for K-concentrated vectors. We extend the result to vectors with indpenent sub-Gaussian entries with an extra logarythmic term. This is a joint work with Nikita Zhivotovskiy. 04.12.18 Yangwen Sun (Humboldt Universität zu Berlin) Complete graph based online change point detection 11.12.18 Mathias Staudigl (Maastricht University) On the convergence of gradient-like flows with noisy gradient input In view of solving convex optimization problems with noisy gradient input, we analyze the asymptotic behavior of gradient-like flows under stochastic disturbances. Specifically, we focus on the widely studied class of mirror descent schemes for convex programs with compact feasible regions, and we examine the dynamics' convergence and concentration properties in the presence of noise. In the vanishing noise limit, we show that the dynamics converge to the solution set of the underlying problem (a.s.). Otherwise, when the noise is persistent, we show that the dynamics are concentrated around interior solutions in the long run, and they converge to boundary solutions that are sufficiently ``sharp". Finally, we show that a suitably rectified variant of the method converges irrespective of the magnitude of the noise (or the structure of the underlying convex program), and we derive an explicit estimate for its rate of convergence. 18.12.18 n. n. 08.01.19 n. n. 15.01.19 Alexander Gasnikov (MIPT) Adaptive accelerated stochastic gradient descent 22.01.19 Ahmed Kebaier (Paris XIII) Seminar at ESH Asymptotic properties of maximum likelihood estimator for the drift parameters of jump-type square root processes 29.01.19 Darina Dvinskikh (WIAS Berlin) Distributed decentralized (stochastic) optimization for dual-friendly functions 05.02.2019 Dr. Jörg Polzehl (WIAS Berlin) Towards in-vivo histology of the brain - some statistical contributions 12.02.19 n. n. Seminar room no. 3.13 at HVP 11a 19.02.2019 n. n. 26.02.19 Dr. Paolo Pigato (WIAS Berlin) Density estimates for stocastic differential equations 05.03.2019 WIAS Day - No Seminar 12.03.19 Malte Weber (HU Berlin) Community detection using belief propagation Using the message passing algorithm "belief propagation" to detect communities in a network graph, i.e. finding subsets of the nodes of the graph which are more densly connected within, than they are to the other nodes of the graph. 19.03.2019 Darina Dvinskikh (WIAS Berlin) Distributed decentralized (stochastic) optimization for dual friendly functions We study the optimal complexity bounds of first-order methods for the problem of minimizing the finite sum of convex (strongly convex) functions. To address this problem, we appeal to distributed optimization and consider the network of multiple agents (processors) which cooperatively minimize the target function. We compare two different approaches toward the finding of the minimum depending on the functions properties in terms of oracle calls and communication rounds: primal approach and dual approach, based on the assumption of existence of the Fenchel-Legendre transform of the target function. 26.03.19 Dr. Valeriy Avanesov (WIAS Berlin) Seminar at ESH Nonparametric change point detection in regression

last reviewed: March 14, 2019 by Christine Schneider