Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Institut für Mathematik

Forschungsseminar Mathematische Statistik

Für den Bereich Statistik

A. Carpentier, S. Greven, W. Härdle, M. Reiß, V. Spokoiny



Weierstrass-Institut für Angewandte Analysis und Stochastik
Mohrenstrasse 39
10117 Berlin



mittwochs, 10.00 - 12.00 Uhr


25. Oktober 2023
Denis Belomestny (Universität Duisburg-Essen)
Provable Benefits of Policy Learning from Human Preferences
Abstract: A crucial task in reinforcement learning (RL) is a reward construction.
It is common in practice that no obvious choice of reward function exists. Thus, a popular approach is to introduce human feedback during training and leverage such feedback to learn a reward function. Among all policy learning methods that use human feedback, preference-based methods have demonstrated substantial success in recent empirical applications such as InstructGPT. In this work, we develop a theory that provably shows the benefits of preference-based methods in tabular and linear MDPs. The main idea of our method is to use KL-regularization with respect to the learned policy to ensure more stable learning.
01. November 2023
Victor Panaretos (EPFL Lausanne)
Optimal transport for covariance operators
Covariance operators are fundamental in functional data analysis, providing the canonical means
to analyse functional variation via the celebrated Karhunen-Loève expansion. These operators may themselves be subject to variation, for instance in contexts where multiple functional populations are to be compared. Statistical techniques to analyse such variation are intimately linked with the choice of metric on covariance operators, and the intrinsic infinite-dimensionality and of these operators. I will describe how the geometry and tools of optimal transportation can be leveraged to construct natural and effective statistical summaries and inference tools for covariance operators, taking full advantage of the nature of their ambient space. Based on joint work with Valentina Masarotto (Leiden), Leonardo Santoro (EPFL), and Yoav Zemel (EPFL).
08. November 2023
Sven Wang (Humboldt-Universität zu Berlin) 
Statistical convergence rates for transport- and ODE-based generative models
Measure transport provides a powerful toolbox for estimation and generative modelling of
complicated probability distributions. The common principle is to learn a transport map which couples a
tractable (e.g. uniform or normal) reference distribution to some complicated target distribution, e.g. by maximizing a likelihood objective. In this talk, we discuss recent advances in statistical convergence
guarantees for such methods. While a general theory is developed, we will primarily treat (1) triangular maps which are the building blocks for “autoregressive normalizing flows" and (2) ODE-based maps, defined through an ODE flow. The latter encompasses NeuralODEs, a popular method for generative modeling. Our results imply that transport methods achieve minimax-optimal convergence rates for non-parametric density estimation over Hölder classes on the unit cube. Joint work with Youssef Marzouk (MIT, United States), Robert Ren (MIT, United States) and Jakob Zech (U Heidelberg, Germany).
15. November 2023
22. November 2023
Marc Hoffmann (Université Paris-Dauphine)
On Estimating Multidimensional Diffusions from Discrete Data

We revisit an old statistical problem: estimate non-parametrically the drift (vector field) and diffusion coefficient (matrix) of a diffusion process from discrete data $(X_0,X_D, X_{2D}, \ldots, X_{ND})$. The novelty are: (i) the multivariate case: only few results have been obtained in this setting from discrete data (and, to the best of our knowledge, no results for the diffusion coefficient) (ii) the sampling scheme has high frequency but is arbitrarily slow: $D=D_N \rightarrow 0$, $ND_N \rightarrow \infty$ and $N_D_N^q\rightarrow 0 from some possibly arbitrarily large $q$ (à la Kessler) and (iii) the process lies in a (not necessarily convex, not necessarily bounded) domain in $\mathbb R^d$ with reflection at the boundary. (In particular we recover the case of a bounded domain or the whole Euclidean space $R^d$.) We establish a relatively standard minimax (adaptive) program for integrated squared error loss over bounded domains (and more losses in the simpler case of the drift) over standard smoothness classes, including lower bounds for the diffusion coefficient with a bit of Malliavin calculus. When $ND_N^2 \rightarrow 0$ and in the special case of the conductivity equation over a bounded domain, we actually obtain contraction rates in squared error loss in a nonparametric Bayes setting. The main difficulty here lies in controlling small ball probabilities for the likelihood ratios; we develop small time expansions of the heat kernel with a bit of Riemannian geometry to control adequate perturbations in KL divergence, using old ideas of Azencott and others. That last part is joint with K.Ray.

Although this problem could have been methodologically addressed almost two decades ago, we heavily rely on the substantial progress that in the domain to clarify and quantify the stability of ergodic averages via concentration chaining techniques and explicit mixing bounds, as well as Malliavin calculus for the lower bound (Dirksen, Gobet, Nickl, Paulin, Ray, Reiss and many others).


29. November 2023

Martin Spindler (Universität Hamburg)

High-Dimensional L2-Boosting: Rate of Convergence  

Boosting is one of the most significant developments in machine learning. This paper studies the rate of convergence of L2Boosting, which is tailored for regression, in a high-dimensional setting. Moreover, we introduce so-called \textquotedblleft post-Boosting\textquotedblright,\textquotedblleft iterated Boosting\textquotedblright and \textquotedblleft restricted Boosting\textquotedblright and \textquotedblleft orthogonal Boosting\textquotedblright and analyze their properties. To show the latter results, we derive new approximation results for the pure greedy algorithm, based on analyzing the revisiting behavior of L2Boosting. We also introduce feasible rules for early stopping, which can be easily implemented and used in applied work. Our results also allow a direct comparison between LASSO and boosting which has been missing from the literature. Finally, we present simulation studies and applications to illustrate the relevance of our theoretical results and to provide insights into the practical aspects of boosting. In these simulation studies, L2Boosting clearly outperforms LASSO.

(joint work with Jannis Kück and Ye Luo)

06. Dezember 2023    
Dennis Nieman (Vrije Universiteit Amsterdam)
Frequentist guarantees for variational Gaussian process regression
We discuss the variational Bayesian approach with inducing variables introduced by Titsias
(2009). This is a sparse approximation for the Bayesian posterior distribution in the nonparametric Gaussian process regression model. The procedure is analyzed from a frequentist perspective: we study contraction rates and validity of the uncertainty quantification. Specifically, it is shown how the frequentist properties of the variational posterior depend on the chosen prior distribution and the dimension of the approximation. Most of the theory is developed under the assumption that the smoothness of the true, data-generating parameter is known, but we also discuss a smoothness-adaptive variational procedure.
13. Dezember 2023    
Boris Buchmann (ANU Canberra)
Weak subordination of multivariate Levy processes
Subordination is the operation which evaluates a Levy process at a subordinator, giving rise
to a pathwise construction of a ``time-changed'' process. In probability semigroups, subordination
was applied to create the variance gamma process, which is prominently used in financial modelling.
However, subordination may not produce a levy process unless the subordinate has independent
components or the subordinate has indistinguishable components. We introduce a new operation
known as weak subordination that always produces a Levy process by assigning the distribution of
the subordinate conditional on the value of the subordinator, and matches traditional subordination in law in the cases above. Weak subordination is applied to extend the class of variance-generalised gamma convolutions and to construct the weak variance-alpha-gamma process.
The latter process exhibits a wider range of dependence than using traditional subordination.
Joint work with Kevin W. LU  - Australian National University (Australia) & Dilip B. Madan - University of Maryland (USA)
15. Dezember 2023    
Laura Sangalli (MOX Milano) 
Physics-Informed Spatial and Functional Data Analysis
Recent years have seen an explosive growth in the recording of increasingly complex and high-
dimensional data, whose analysis calls for the definition of new methods, merging ideas and approaches from statistics and applied mathematics. My talk will focus on spatial and functional data observed over non-Euclidean domains, such as linear networks, two-dimensional manifolds and non-convex volumes. I will present an innovative class of methods, based on regularizing terms involving Partial Differential Equations (PDEs), defined over the complex domains being considered. These Physics--Informed statistical learning methods enable the inclusion of the available problem specific information, suitably encoded in the regularizing PDE. Illustrative applications from environmental and life sciences will be presented.
20. Dezember 2023    
10. Januar 2024
Eric Moulines (Ecolé Polytechnique)
Score-based diffusion models and applications
Deep generative models represent an advanced frontier in machine learning. These models are adept at fitting complex data sets, whether they consist of images, text or other forms of high-dimensional data. What makes them particularly noteworthy is their ability to provide independent samples from these complicated distributions at a cost that is both computationally efficient and resource efficient. However, the task of accurately sampling a target distribution presents significant challenges. These challenges often arise from the high dimensionality, multimodality or a combination of these factors. This complexity can compromise the effectiveness of traditional sampling methods and make the process either computationally prohibitive or less accurate.In my talk, I will address recent efforts in this area that aim to improve traditional inference and sampling algorithms. My major focus will be on score-based diffusion models. By utilizing the concept of score matching and time-reversal of stochastic differential equations, they offer a novel and powerful approach to generating high-quality samples. I will discuss how these models work, their underlying principles and how they are used to overcome the limitations of conventional methods. The talk will also cover practical applications, demonstrating their versatility and effectiveness in solving complex real-world problems.
17. Januar 2024
Matteo Giordano (University of Torino)
Likelihood Methods for Low Frequency Diffusion Data
The talk will consider the problem of nonparametric inference in multi-dimensional diffusion models from low-frequency data. Implementation of likelihood-based procedures in such settings is a notoriously delicate task, due to the computational intractability of the likelihood. For the nonlinear inverse problem of inferring the diffusivity in a stochastic differential equation, we propose to exploit the underlying PDE characterisation of the transition densities, which allows the numerical evaluation of the likelihood via standard numerical methods for elliptic eigenvalue problems. A simple Metropolis-Hastings-type MCMC algorithm for Bayesian inference on the diffusivity is then constructed, based on Gaussian process priors. Furthermore, the PDE approach also yields a convenient characterisation of the gradient of the likelihood via perturbation techniques for parabolic PDEs, allowing the construction of gradient-based inference methods including MLE and Langevin-type MCMC. The performance of the algorithms is illustrated via the results of numerical experiments. Joint work with Sven Wang.
24. Januar 2024
Simon Wood (The University of Edinburgh)
On Neighbourhood Cross Validation

Cross validation comes in many varieties, but some of the more interesting flavours require multiple model fits with consequently high cost. This talk shows how the high cost can be side-stepped for a wide range of models estimated using a quadratically penalized smooth loss, with rather low approximation error.

Once the computational cost has the same leading order as a single model fit, it becomes feasible to efficiently optimize the chosen cross-validation criterion with respect to multiple smoothing/precision parameters. Interesting applications include cross-validating smooth additive quantile regression models, and the use of leave-out-neighbourhood cross validation for dealing with nuisance short range autocorrelation.

The link between cross validation and the jackknife can be exploited to  obtain reasonably well calibrated uncertainty quantification in these cases.

31. Januar 2024
Gianluca Finocchio (Universität Wien)
An extended latent factor framework for ill-posed linear regression
The classical latent factor model for linear regression is extended by assuming that, up to an unknown orthogonal transformation, the features consist of subsets that are relevant and irrelevant for the response. Furthermore, a joint low-dimensionality is imposed only on the relevant features vector and the response variable. This framework allows for a comprehensive study of the partial-least-squares (PLS) algorithm under random design. In particular, a novel perturbation bound for PLS solutions is proven and the high-probability L²-estimation rate for the PLS estimator is obtained. This novel framework also sheds light on the performance of other regularisation methods for ill-posed linear regression that exploit sparsity or unsupervised projection. The theoretical findings are confirmed by numerical studies on both real and simulated data.
07. Februar 2024
Evgenii Chzhen (LMO Orsay, Paris)
Small Total-Cost Constraints in Contextual Bandits with Knapsacks
I will talk about some recent developments in the literature of contextual bandit problems with knapsacks [CBwK], a problem where at each round, a scalar reward is obtained and vector-valued costs are suffered. The goal is to maximize the cumulative rewards while ensuring that the cumulative costs are lower than some predetermined cost constraints. In this setting, total cost constraints had so far to be at least of order T^{3/4} where T is the number of rounds, and were even typically assumed to depend linearly on T. Elaborating on the main technical challenge and drawback of the previous approaches, I will present a dual strategy based on projected-gradient-descent updates, that is able to deal with total-cost constraints of the order of T^{1/2} up to poly-logarithmic terms. This strategy is direct, and it relies on a careful, adaptive, tuning of the step size. The approach is inspired by a parameter-free-type algorithms arising from convex (online) optimization literature, which I also briefly review.
The talk is based on joint works with C. Giraud, Z. Li, and G. Stoltz
14. Februar 2024
Martin Wahl (U Bielefeld)
Heat kernel PCA with applications to Laplacian Eigenmaps

Laplacian eigenmaps and diffusion maps are nonlinear dimensionality reduction methods that use the eigenvalues and eigenvectors of (un)normalized graph Laplacians. Both methods are applied when the data is sampled from a low-dimensional manifold, embedded in a high-dimensional Euclidean space. From a mathematical perspective, the main problem is to understand these empirical Laplacians as spectral approximations of the underlying Laplace-Beltrami operator. In this talk, we study Laplacian eigenmaps through the lens of kernel PCA, and consider the heat kernel as reproducing kernel feature map. This leads to novel points of view and allows to leverage results for empirical covariance operators in infinite dimensions.


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Frau Sabine Bergmann

Telefon: +49-30-2093-45450
Fax:        +49-30-2093-45451
Humboldt-Universität zu Berlin
Institut für Mathematik
Unter den Linden 6
10099 Berlin, Germany