Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Andrea Fiebig

Berlin Probability Colloquium

 

 

Ort

HU Berlin, Institut für Mathematik
Rudower Chaussee 25, Raum 1.115
12489 Berlin

 

Zeit

mittwochs, 16.15 - 18.00  Uhr

Programm

 

26. Oktober 2022
Raul Tempone (KAUST & RWTH Aachen) -- 16.15 Uhr
Multi-iteration Estimators for Stochastic Optimization
Abstract: We will discuss using adaptive control variates for mean gradient estimation along the iterates produced by a generic stochastic optimization algorithm. We couple the resulting Multi-Iteration Estimator (MICE) with several first-order stochastic optimizers and show efficiency gains in several examples. We provide theoretical results for the strongly convex case, where the control of the coefficient of variation of the mean gradient approximation plays a crucial role. To assess the usefulness of MICE, we present several examples in which we use SGD-MICE and Adam-MICE. We include one example based on a stochastic adaptation of the Rosenbrock function and logistic regression training for various datasets. When compared to SGD, SAG, SAGA, SVRG, and SARAH methods, the Multi-Iteration Stochastic Optimizers reduced, without the need to tune parameters for each example, the gradient sampling cost in all cases tested, also lowering the total runtime in some cases. Finally, if time allows, we will briefly introduce Quasi-Newton preconditioners based on noisy gradient observations and discuss their use to improve convergence in examples with large condition numbers.
References:
• “Multi-Iteration Stochastic Optimizers”, by A.Carlon, L.Espath, R.Lopez and R.Tempone. arXiv:2011.01718, November 2020.
• “Approximating Hessian matrices using Bayesian inference: a new approach for quasi-Newton methods in stochastic optimization,” by A. Carlon, L. L. Espath and R. Tempone. arXiv:2208.00441, July 2022.
Nils Berglund (Université d’Orléans) -- 17.15 Uhr
Stochastic resonance in stochastic PDEs
Abstract: Stochastic resonance can occur when a multi-stable system is subject to both periodic and random perturbations. For suitable parameter values, the system can respond to the perturbations in a way that is close to periodic. This phenomenon was initially proposed as an explanation for glacial cycles in the Earth’s climate. While its role in that context remains controversial, stochastic resonance has since been observed in many physical and biological systems. This talk will focus on stochastic resonance in parabolic SPDEs, such as the Allen-Cahn equation, when they are driven by a periodic perturbation and by space-time white noise. We will discuss both the case of one spatial dimension, in which the equation is well-posed, and the case of two spatial dimensions, in which a renormalisation procedure is required. This talk is based on joint works with Barbara Gentz and Rita Nader. References:
• https://dx.doi.org/10.1007/s40072-021-00230-w
• https://arxiv.org/abs/2107.07292.
• https://arxiv.org/abs/2209.15357
09. November 2022
Hanno Gottschalk (Bergische Univ. Wuppertal) -- 16.15 Uhr
Foundations and Applications of Generative Adversarial Learning
Abstract: Generative Learning is the task of learning from data, how to sample from a high dimensional probability distribution. One prominent example is generative adversarial learning, where a discriminator is trained to distinguish between real and synthesized samples whereas a generator is trained to make this task for the discriminator as hard as possible. In this way, using neural networks for generation and discrimination, stunning results have been obtained, e.g. in synthesizing images of persons who never existed.
In this talk, an introduction to the theory of generative adversarial learning is given, where generators and discriminators are Hölder functions. It is proven that generative learning in this infinite dimensional setting is consistent, and furthermore rates of convergence for the Jensen-Shannon divergence of the real and the synthetic measure are given.
Numerical results for generative adversarial learning with the application to domain adaptation and the simulation of turbulent flows are given as well.
Jochen Blath (Goethe-Universität Frankfurt) -- 17.15 Uhr
The effects of dormancy in population genetics, evolution and ecology
Abstract: Throughout the tree of life populations have evolved the capacity to contend with suboptimal conditions by engaging in dormancy, whereby individuals enter a reversible state of vanishing metabolic activity. The resulting “seed banks” serve as long-lived reservoirs of genotypic and phenotypic diversity. Of particular relevance is the case of microbial dormancy, which has a fundamental impact on the evolutionary, ecological and also pathogenic character of biological communities.
However, despite its ubiquity in nature, dormancy is a rather new paradigm in stochastic interacting particle systems. Here, it leads to novel effects, in particular based on the introduction of memory, resilience and diversity into the underlying systems. The resulting probabilistic structures, arising on different scales, are surprisingly rich, already when considering simple ‘toy models’, and lead to new universal behaviour.
In this talk, after providing some background on dormancy, I will present and discuss recent models for dormancy and seed banks in population genetics, evolution and ecology. Along the line, I will sketch topics for future mathematical and interdisciplinary research.
23. November 2022
 Dylan Possamai (ETH Zurich) -- 16.15 Uhr
Moral hazard for time-inconsistent agents and BSVIEs
Abstract: We address the problem of Moral Hazard in continuous time between a Principal and an Agent that has time-inconsistent preferences. Building upon previous results on non-Markovian time-inconsistent control for sophisticated agents, we are able to reduce the problem of the principal to a novel class of control problems, whose structure is intimately linked to the representation of the problem of the Agent via a so-called extended Backward Stochastic Volterra Integral equation. We will present some results on the characterization of the solution to problem for different specifications of preferences for both the Principal and the Agent.
Huyen Pham (Université de Paris) -- 17.15 Uhr
Learning in continuous time mean-field control problems
Abstract: The theory and applications of mean-field games/control have stimulated a growing interest and generated important literature over the last decade since the seminal papers by Lasry/Lions and Caines, Huang, Malhamé. This talk will address some learning methods for numerically solving continuous time mean-field control problems, also called McKean-Vlasov control (MKV). In a first part, we consider a model-based setting, and present numerical approximation methods for the Master Bellman equation that characterises the solution to MKV, based on the one hand on particles approximation of the Master equation, and on the other hand on cylindrical neural networks approximation of functions defined on the Wasserstein space. The second part of the lecture is devoted to a model-free setting, a.k.a. reinforcement learning. We develop a policy gradient approach under entropy regularisation based on a suitable representation of the gradient value function with respect to parametrised randomised policies. This study leads to actor-critic algorithms for learning simultaneously and alternately value function and optimal policies. Numerical examples in a linear-quadratic mean-field setting illustrate our results.
07. Dezember 2022
Xiaolu Tan (Hong Kong) -- 16.15 Uhr
A mean-field version of Bank–El Karoui's representation of stochastic processes''
Abstract: We study a mean-field version of Bank–El Karoui’s representation theorem of stochastic processes. Under different technical conditions, we establish some existence and uniqueness results. In particular, we derive a stability result on the classical representation, which would have its own interests. Finally, as motivation and applications, our mean-field representation results provide a unified approach to study different Mean-Field Games (MFGs) in the setting with common noise and multiple populations, including the MFG of timing, the MFG with singular control, etc.
Wei Xu (HU Berlin) -- 17.15 Uhr
Functional limit theorems for quasi-stationary Hawkes processes
Abstract: n this talk, we introduce several functional law of large numbers (FLLN) and functional central limit theorem (FCLT) for quasi-stationary Hawkes processes. Under some divergence conditions on triggered events, we prove that the normalized point processes can be approximated in distribution by a long-range dependent Gaussian process. Differently, both FLLN and FCLT fail when triggered events satisfy aggregation conditions. In this case, we prove that the rescaled Hawkes process converges weakly to the integral of a critical branching diffusion with immigration. Also, the convergence rate is deduced in terms of Fourier-Laplace distance bound and Wasserstein distance bound. This talk is based on a joint work with Ulrich Horst.
11. Januar 2023
Jan Palczewski (U Leeds) -- 16.15 Uhr
Equilibria in non-Markovian zero-sum stopping games with asymmetric information
Abstract: I will show that a zero-sum stopping game in continuous time with partial and/or asymmetric information admits a saddle point (and, consequently, a value) in randomised stopping times when stopping payoffs of players are general càdlàg adapted processes. We do not assume a Markovian nature of the game nor a particular structure of the information available to the players. I will discuss links with classical results by Baxter, Chacon (1977) and Meyer (1978) derived for optimal stopping problems. Based on a joint work with Tiziano De Angelis, Nikita Merkulov and Jacob Smith.
Chiranjib Mukherjee (WWU Münster) -- 17.15 Uhr
Subcritical Gaussian multiplicative chaos in the Wiener space
Abstract: We construct and study properties of an infinite dimensional analog of Kahane’s theory of Gaussian multiplicative chaos (GMC). Namely, we consider a random field defined with respect to space-time white noise integrated w.r.t. Brownian paths in d ≥ 3 and construct the infinite volume limit of the renormalized exponential of this field, weighted w.r.t. the Wiener measure, in the entire weak disorder (subcritical) regime. We show that this subcritical GMC is unique w.r.t. the mollification scheme in the sense of Shamov and determine its support by identifying its thick points, implying the singularity of this object. We also prove, in the subcritical regime, existence of negative and positive (Lp for p > 1) moments of the total mass, and deduce its Hölder exponents (small ball probabilities) explicitly. Joint work with Rodrigo Bazaes (Münster) and Isabel Lammers (Münster).
25. Januar 2023
Lukas Gonon  (Imperial College) -- 16.15 Uhr
Detecting asset price bubbles using deep learning
Abstract: In this talk we present a novel deep learning methodology to detect financial asset bubbles by using observed call option prices. We start with an introduction to deep learning and asset price bubbles. We then examine the pitfalls of a naive approach for deep learning-based bubble detection and subsequently introduce our method. We provide theoretical foundations for the method in the context of local volatility models and show numerical results from experiments both on simulated and market data. The talk is based on joint work with Francesca Biagini, Andrea Mazzon and Thilo Meyer-Brandis.
Avi Mayorcas (TU Berlin) -- 17.15 Uhr
Blow-up criteria for an SPDE model of chemotaxis
Abstract: Chemotaxis and related phenomena have been an active area of mathematical research since statistical and PDE models were first proposed by C. Patlak (’53) and E. Keller & L. Segel (’71). They are commonly studied through mean field PDE models and a common feature of these equations is the possibility of finite time blow-up under given model parameters. Recently it was shown that advection by a sufficiently strong relaxation enhancing vector field could suppress this blow up (Kiselev & Xu ’16, Iyer, Zlatos & Xu ’20). In this talk I will discuss new results (obtained with M. Tomašević) regarding criteria for the persistence of blow-up for an SPDE model of chemotaxis with stochastic advection. The noise we cover is of a form recently shown to be almost surely relaxation enhancing (Gess & Yaroslavtsev ’21) and closely related to those studied in recent works by Galeati, Flandoli and Luo.
15. Februar 2023
Mathias Trabs  (KIT) -- 16.15 Uhr
Statistics for SPDEs based on discrete observations
Abstract: We study parameter estimation for stochastic PDEs when the solution field is observed discretely in time and space. In one space dimension we prove central limit theorems for realized quadratic variations based on temporal and spatial increments as well as on rectangular increments in time and space assuming an infill asymptotic regime in both coordinates. Moreover, we aim for bridging the gap between the discrete observation scheme and the local measurements approach. In particular, the discrete Laplacian understood as a local measurement of distribution type will allow for analyzing estimators based on discrete observations in arbitrary space dimensions. Joint works with Randolf Altmeyer, Markus Bibinger and Florian Hildebrandt 
Ana Djurdjevac  (FU Berlin) -- 17.15 Uhr
Synchronisation for scalar conservation laws via Dirichlet boundary 
Abstract: In the talk we will discuss long-time properties of stochastic scalar conservation laws with homogenous Dirichlet conditions. As a motivation of the work, we will briefly discuss stabilisation of a viscous conservation law by a one-dimensional external force. The natural idea of proving the synchronization is to separate the boundary effect from the bulk behavior. This approach requires the presence of a non-degenerate force and the solution of a control problem. In the present work we take a different approach, and provide a bound on the contraction constant through an elementary estimate on the dissipation of mass at the boundary, which does not require any non-degeneracy condition. We identify a coercivity condition under which the estimates are uniform over all initial conditions, via the construction of suitable super- and sub-solutions. In lack of such coercivity our results build on L^p energy estimates and a Lyapunov structure. This is a joint with T. Rosati.