Exploring Uncertainty: Stochastic Systems, Risk Management, and Machine Learning

Navier-Stokes equations in dimension d>2 are considered. A generalisation of well-known results by Prodi and Serrin on conditional uniqueness of its solution is presented, and it is extended also to stochastic Navier-Stokes equations. The talk is based on a recent joint work with Nicolai V. Krylov.

For a given finite set of random variables of finite value, such as categories, the loglinear model for their distribution is characterized by the property that the joint distribution can be written in form of a product, where each factor depends only on the values of the random variables belonging to some prescribed subsets, called defining subsets. For a given empirical distribution, the maximum likelihood (ML) estimate may not exist. A necessary condition is that the marginals of the empirical distribution corresponding to the defining subsets should have full support. It is well-known that if the loglinear model is decomposable, the ML estimate exists under the previous necessary condition. Moreover, in this case, there is an explicit formula giving the ML estimate from its defining marginals. In the present talk, we address the problem of whether the decomposable property can be weakened, in other words, to characterize the loglinear models for which the previous necessary condition for the existence of the ML estimate is also sufficient. We show by constructing counterexamples that only the decomposable loglinear models have this property. Joint work with Balázs Gerencsér.

We consider here an application of Fuhrmann's Reachability Map to Tangential Interpolation problems. We construct the J-inner function Θ(s) which is used to parametrise all solutions directly from the data, without using RKHS or Lyapunov--Stein Equations, tools which are essential to the existing approaches and involve either the Schur algorithtm (which requires, as we show, an unnecessary renormalisation) or the inversion of a Pick matrix. The technique yields, in general, an LU factorisation of several notable matrices, like Toeplitz, Nevanlinna-Pick, Sylvester, Bezout matrices). More generally, we show that, if a full-rank matrix P satisfies a Lyapunov-Stein equation of the form AP + PÂ+WŴ=0 (with A and  lower and upper triangular) and admits an LU factorisation, this factorisation can be computed without actually solving the equation. Joint work with Gy. Michaletzky.

Optimal experimental designs were discussed first in the statistical literatures as allocation problem of points in classical regression problems. This was soon followed by designing optimal input or excitation signals under special constraints to get a kind of best estimation of parametrized models from measured data. These procedures were very useful to get the minimal number of experiments to achieve a certain quality of the estimated parameters or frequency response. Optimality of the designs were formulated both in time and in the frequency domain and various off – line and sequential algorithms were developed. Optimal experiment design (OED) approaches were applied in medical experiments, control of industrial processes, identification and control of vehicle dynamics, and recently there is an emerging interest in applying it in machine learning under the name active learning, for problems such as classification. This talk will survey the basic problems in regression experimental design with an extension to linear and nonlinear models for dynamic systems. Some recent results on frequency domain multi-sine design will be presented, too, that leads to a sparse estimation problem of the multi-sine frequencies (and amplitudes) yielding a sub-optimally estimated model. Similar tools were used to get approximate dynamics of vehicles and space structures. Joint work with Zoltán Szabó.

Stochastic multi-armed bandits (MABs) provide a fundamental framework to study sequential decision making in uncertain environments. The upper confidence bounds (UCB) algorithm is a primary choice to solve these problems as it achieves near-optimal regret rates under various moment assumptions. Up until recently most UCB methods relied on concentration inequalities leading to confidence bounds which depend on moment parameters, such as the variance proxy of subgaussian distributions, that are usually unknown in practice. In this talk we present a new distribution-free, data-driven UCB algorithm for symmetric reward distributions which is completely parameter-free, e.g., it needs no moment information. The key idea is to combine a refined, one-sided version of the recently developed resampled median-of-means (RMM) estimator with UCB. The resulting anytime, parameter-free RMM-UCB algorithm achieves near-optimal regret, even for heavy-tailed reward distributions. Experiments also show that RMM-UCB outperforms most state-of-the-art bandit algorithms on difficult MAB problems, i.e., when the suboptimality gap is small and the reward distributions are heavy-tailed. Joint work with Ambrus Tamás and Szabolcs Szentpéteri.

We overview recent results on assessing the convergence rate of average consensus algorithms. Besides symmetric linear updates, a popular communication protocol in focus is push-sum (or ratio consensus), and our goal is to discuss their almost sure exponential convergence rate. We present multiple ways to address this problem, balancing precision and scalability. Joint work with L. Gerencsér, M. Kornyik.

We review recent results about Markov chains whose transition kernel depends also on a given stationary process, called the environment. We concentrate on models from two application areas: stochastic volatility in financial mathematics and optimization algorithms in machine learning with dependent data. We present results on convergence to an equilibrium state and on mixing properties of the processes involved.