Modeling Nonstationary Extremal Dependence via Deep Spatial Deformations

Modeling nonstationarity that often prevails in extremal dependence of spatial data can be challenging, and typically requires bespoke or complex spatial models that are difficult to estimate. Inference for stationary and isotropic models is considerably easier, but the assumptions that underpin these models are rarely met by data observed over large or topographically complex domains. A possible approach for accommodating nonstationarity in a spatial model is to warp the spatial domain to a latent space where stationarity and isotropy can be reasonably assumed. Although this approach is very flexible, estimating the warping function can be computationally expensive, and the transformation is not always guaranteed to be bijective, which may lead to physically unrealistic transformations when the domain folds onto itself. We overcome these challenges by developing deep compositional spatial models to capture nonstationarity in extremal dependence. Specifically, we focus on modeling high threshold exceedances of process functionals by leveraging efficient inference methods for limiting $r$-Pareto processes. A detailed high-dimensional simulation study demonstrates the superior performance of our model in estimating the warped space. We illustrate our method by modeling UK precipitation extremes and show that we can efficiently estimate the extremal dependence structure of data observed at thousands of locations.

Multi-Agent Q-Learning Dynamics in Random Networks: Convergence due to Exploration and Sparsity

Beyond specific settings, many multi-agent learning algorithms fail to converge to an equilibrium solution, and instead display complex, non-stationary behaviours such as recurrent or chaotic orbits. In fact, recent literature suggests that such complex behaviours are likely to occur when the number of agents increases. In this paper, we study Q-learning dynamics in network polymatrix games where the network structure is drawn from classical random graph models. In particular, we focus on the Erdos-Renyi model, a well-studied model for social networks, and the Stochastic Block model, which generalizes the above by accounting for community structures within the network. In each setting, we establish sufficient conditions under which the agents' joint strategies converge to a unique equilibrium. We investigate how this condition depends on the exploration rates, payoff matrices and, crucially, the sparsity of the network. Finally, we validate our theoretical findings through numerical simulations and demonstrate that convergence can be reliably achieved in many-agent systems, provided network sparsity is controlled.