Conference Agenda

Beckmann’s problem in optimal transport minimizes the total squared flux in a continuous transport problem from a source to a target distribution. In this article, the regularity theory for solutions to Beckmann’s problem in optimal transport is developed utilizing an unconstrained Lagrangian formulation and solving the variational first order optimality conditions. It turns out that the Lagrangian multiplier that enforces Beckmann’s divergence constraint fulfills a Poisson equation and the flux vector field is obtained as the potential’s gradient. Utilizing Schauder estimates from elliptic regularity theory, the exact Hölder regularity of the potential, the flux and the flow generating is derived on the basis of Hölder regularity of source and target densities on a bounded, regular domain. If the target distribution depends on parameters, as is the case in conditional (“promptable”) generative learning, we provide sufficient conditions for separate and joint Hölder continuity of the resulting vector field in the parameter and the data dimension. Following a recent result by Belomnestny et al., one can thus approximate such vector fields with deep ReQu neural networks in C^k,α -Hölder norm. We also show that this approach generalizes to other probability paths, like Fisher-Rao gradient flows.

Given data generated by an observable stochastic process, we study how to construct statistically optimal decisions for general stochastic optimization problems. Our setting encompasses non-standard data structures, including data originating from heterogeneous sources or from randomly evolving data-generating mechanisms. We propose a decision-making approach that identifies optimal decisions for which a specific notion of risk of shifted regret decays to zero at a prescribed exponential rate. This optimal decision arises as the solution to a multi-objective optimization problem, which reflects asymptotic behavior properties of the data-generating process. Central to our framework is a rate function that characterizes this behavior via a Laplace principle, thereby generalizing standard concepts from large deviation theory. Our general formulation enables our approach to account for data from uncertain distributions and recovers classical results in data-driven decision making under uncertainty as special cases, including distributionally robust optimization. Moreover, our method enables decision-makers to systematically balance a desired rate of asymptotic risk decay against a potential loss in statistical consistency of the resulting data-driven decision. We demonstrate the effectiveness of the proposed approach through illustrative examples from operations research, such as the newsvendor problem, under aleatoric uncertainty induced by heterogeneous data sources.

Random Weights Neural Networks (RWNNs) have attracted significant attention in the literature and practical applications due to their ability to approximate complex functions with minimal and easy-to-implement training. Although prior research has established the universal approximation properties and generalization bounds of these methods, statistical inference has remained largely unexamined. Leveraging recent advances in approximation theory, we derive faster-than-standard Monte Carlo approximation rates in both L2 and expected supremum norms. We then establish asymptotic consistency and normality of RWNN estimators in L2 and uniform consistency under appropriate conditions. Our results highlight the interaction between randomness, regularization, and conditioning in determining the quality of the inference.