PAC Bayesian bounds for light cone, neighborhood, and trajectories data

• Applicant: Imma Valentina Curato, Ph.D.
• Subject Area: Mathematics
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 467779727

Light cone, neighborhood, and trajectories data are important types of spatio-temporal data sets analyzed, for example, in climate and earth science, medicine, economics, and intelligent transportation. Determining supervised learning algorithms that consider the characteristic serial correlation of such spatio-temporal data sets is an open question of utmost importance for the forecasting ability of a proposed algorithm. The project aims to develop a mathematical foundation of supervised learning algorithms for spatio-temporal data via the determination of PAC Bayesian bounds. First, the latter is analyzed by assuming that the observed data are identically distributed, and $\theta$-lex or $\eta$-weakly dependent. These notions seem pivotal for determining bounds for the generalization error of a learning task, assuming that spatio-temporal random fields generate the data. Thus, the shape of PAC Bayesian bounds for broad classes of stationary random fields, namely, mixed moving average and ambit fields, is examined, and the tightness of the discovered bounds to the generalization error is investigated. This analysis should give the basis to set-up a model selection strategy for linear predictors applied to light cone, neighborhood, and trajectories data.Second, a general spatio-temporal data set contaminated by noise is considered without restricting its generation. In this set-up, the PAC Bayesian bound to be developed applies to nonparametric regression models. This bound should allow determining a model selection strategy for learning Gaussian processes.In the current literature, the PAC Bayesian bounds are obtained for bounded and unbounded loss functions and typically hold when asking for finite exponential moments, which is a quite restrictive assumption in the spatio-temporal framework. In this project, the PAC Bayesian bounds are determined for unbounded and locally Lipschitz functions simply assuming finite second moments.

• DFG Programme: Research Grants