Many sequential decision problems are problems of optimal stopping. When decision items are independent of each other, then the so-called odds-algorithm plays a distinguished role: It gives online the optimal stopping time as well as the corresponding optimal value simultaneously. Moreover it is optimal itself in the sense that no other decision algorithm can do this more quickly. These combined properties, rare among methods to solve decision problems, are the reason why modifications of the odds-algorithm, based on learning in the non-independent case, should attract particular interest. This talk presents new results on what we could call "moving odds-algorithms" which we think of being well-suited for real-world applications.

We consider a population model in which the season alternates between winter and summer. Individuals can acquire mutations that are advantageous in the summer but disadvantageous in the winter, or vice versa. Furthermore, it is assumed that individuals within the population can either be active or dormant, and that individuals can transition between these two states. Dormant individuals do not reproduce and are not subject to selective pressures. Our findings indicate that, under some conditions, two waves of adaptation emerge over time. Some individuals repeatedly acquire mutations that are beneficial in the summer, while others repeatedly acquire mutations that are beneficial in the winter. Individuals can survive the season during which they are less fit by entering a dormant state. This result demonstrates that, for populations in fluctuating environments, dormancy has the potential to induce speciation.

In this talk, we examine the use of metaheuristic methods based on large systems of stochastic interacting particles to solve complex optimization problems, focusing on the Particle Swarm Optimization (PSO) method. PSO is inspired by collective intelligence, where particles adjust their positions based on their own performance and the influence of their neighbors, while incorporating stochastic exploration to enhance the search for optimal solutions. We will discuss the convergence properties of PSO to global minimizers and its connection to Consensus-Based Optimization (CBO) in the limiting case of zero inertia.

We want to examine the weekly spatio-temporal distribution of mortalities in Austria over the time period 2002-2023 and investigate to which extend this distribution can be explained by meteorological variables as weekly maximum, minimum and mean temperature, weekly length of a dry spell, (lagged) heat and cold waves, weekly mean humidity and mean precipitation, the interaction between minimum or maximum temperature and humidity, but also by age, gender and population. Therefore, we set up a model for mortality rates that includes space-time, space-age, and age-time interactions.

We consider two different ways to compare probability measures: optimal transportation cost (or Wasserstein distance) on the one hand, and relative entropy (or Küllback-Leibler divergence) on the other. An inequality between these quantities holds, for example, when the reference measure is a standard Gaussian on Euclidean space, or the law of a Brownian motion. We lift this inequality to Wiener measure on the space of rough paths, i.e. the law of Brownian motion taken together with its Lévy area. We present two approaches to the proof that reveal phenomena not present in the case of classical Wiener space. Joint work with Peter Friz, Helena Kremp, Vaios Laschos, and Matthias Liero.

We consider the framework of penalized estimation where the penalty term is given by a polyhedral norm, or more generally, a polyhedral gauge, which encompasses methods such as LASSO and generalized LASSO, SLOPE, OSCAR, PACS and others. Each of these estimators can uncover a different structure or “pattern” of the unknown parameter vector. We define a novel and general notion of patterns based on subdifferentials and formalize an approach to measure pattern complexity. For pattern recovery, we provide a minimal condition for a particular pattern to be detected with positive probability, the so-called accessibility condition. We make the connection to estimation uniqueness by showing that uniqueness holds if and only if no pattern with complexity exceeding the rank of the X-matrix is accessible. Subsequently, we introduce the noiseless recovery condition which is a stronger requirement than accessibility and which can be shown to play exactly the same role as the well-known irrepresentability condition for the LASSO – in that the probability of pattern recovery is bounded by 1/2 if the condition is not satisfied. Through this, we unify and extend the irrepresentability condition to a broad class of penalized estimators using an interpretable criterion. We also look at the “gap” between accessibility and the noiseless recovery condition and discuss that our criteria show that it is more pronounced for simple patterns. Finally, we prove that the noiseless recovery condition can indeed be relaxed when turning to so-called thresholded penalized estimation: in this setting, the accessibility condition is already sufficient (and necessary) for sure pattern recovery provided that the signal of the pattern is large enough. We demonstrate how our findings can be interpreted through a geometrical lens throughout the talk and illustrate our results for the Lasso as well as other estimation procedures.

In this talk, the approximation of jump-diffusion stochastic differential equations with discontinuous drift, possibly degenerate diffusion coefficient, and Lipschitz continuous jump coefficient is studied. These stochastic differential equations can be approximated with a jump-adapted approximation scheme with a convergence rate 3/4 in $L^p$. This rate is optimal for jump-adapted approximation schemes. We present an advanced adaptive approximation scheme to improve this convergence rate. Our scheme obtains a strong convergence rate of at least 1 in $L^p$ in terms of the average number of evaluations of the driving noises.

There have been a lot of recent progress on branching particle systems with selection, in particular on the $N$-particle branching random walk ($N$-BRW). In the $N$-BRW, $N$ particles have locations on the real line at all times. At each time step, every particle generates a number of children, and each child has a random displacement from its parent's location. Then among the children only the $N$ rightmost are selected to survive and reproduce in the next generation. In this talk we will investigate a noisy version of the $N$-BRW. In this model the $N$ surviving particles are selected at random from the children in such a way, that particles more to the right on the real line are more likely to be selected. I will present some recent results on the asymptotic behaviour of our particle system as $N$ goes to infinity; including the distribution of the $N$ particles on the real line and the genealogical properties of the system. Our results show that as we change the selection parameter, there is a phase transition in these asymptotic properties. This is joint work with Colin Desmarais, Bastien Mallein, Francesco Paparella and Emmanuel Schertzer.