Forschungsgruppe "Stochastische Algorithmen und Nichtparametrische Statistik"
Research Seminar "Mathematical Statistics" Winter Semester 2023/24
|
|
| 18.10.2023 | N.N. |
| |
|
| 25.10.2023 | Prof. Dr. Denis Belomestny (Universität Duisburg-Essen) |
| Provable benefits of policy learning from human preferences 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. |
|
| 01.11.2023 | Prof. Dr. 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). |
|
| 08.11.2023 | Dr. 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). |
| 15.11.2023 | N. N. |
| |
|
| 22.11.2023 | Prof. Dr. Marc Hoffmann (Université Paris-Dauphine) |
| Achtung anderer Raum: 406, 4. OG ! | On estimating multidimensional diffusions from discrete data
|
| 29.11.2023 | Prof. Dr. Martin Spindler (Universität Hamburg) |
| Achtung anderer Raum u. anderes Geb.: R. 3.13 im HVP 11a ! | High-dimensional L2-boosting: Rate of convergence (hybrid talk) |
| 06.12.2023 | N.N. |
| |
|
| 13.12.2023 | Dr. Boris Buchmann (ANU Canberra, Australia) |
| 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) |
|
| Freitag, 15.12.2023 | Prof. Dr. Laura Sangalli (MOX Milano, Italien) |
| An additional session! | 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. |
| 20.12.2023 | N.N. |
|
|
|
| 10.01.2024 | Prof. Dr. Eric Moulines (Ecole 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. |
|
| 17.01.2024 | Dr. Matteo Giordano (Università degli Studi di 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. |
|
| 24.01.2024 | Professsor Simon Wood (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 |
|
| 31.01.2024 | PhD Gianluca Finocchio (Universität Wien) |
| Achtung anderer Raum: 406, 4. OG ! | 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. |
| 07.02.2024 | PhD Evgenii Chzhen (LMO Orsay, Paris) | Small total-cost constraints in contextual bandits with knapsacks |
| 14.02.2024 | Prof. Dr. Martin Wahl (Universität 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. |
last reviewed: March 25, 2024 by Christine Schneider