PAC learning with generalized samples and an application to stochastic geometry

Includes bibliographical references (p. 16-17).Caption title.Research supported by the National Science Foundation. ECS-8552419 Research supported by the U.S. Army Research Office. DAAL01-86-K-0171 Research supported by the Dept. of the Navy under an Air Force Contract. F19628-90-C-0002S.R. Kulkarni ... [et al.]

Probably Approximately Correct Nash Equilibrium Learning

We consider a multi-agent noncooperative game with agents' objective functions being affected by uncertainty. Following a data driven paradigm, we represent uncertainty by means of scenarios and seek a robust Nash equilibrium solution. We treat the Nash equilibrium computation problem within the realm of probably approximately correct (PAC) learning. Building upon recent developments in scenario-based optimization, we accompany the computed Nash equilibrium with a priori and a posteriori probabilistic robustness certificates, providing confidence that the computed equilibrium remains unaffected (in probabilistic terms) when a new uncertainty realization is encountered. For a wide class of games, we also show that the computation of the so called compression set - a key concept in scenario-based optimization - can be directly obtained as a byproduct of the proposed solution methodology. Finally, we illustrate how to overcome differentiability issues, arising due to the introduction of scenarios, and compute a Nash equilibrium solution in a decentralized manner. We demonstrate the efficacy of the proposed approach on an electric vehicle charging control problem.Comment: Preprint submitted to IEEE Transactions on Automatic Contro

Changes from Classical Statistics to Modern Statistics and Data Science

A coordinate system is a foundation for every quantitative science, engineering, and medicine. Classical physics and statistics are based on the Cartesian coordinate system. The classical probability and hypothesis testing theory can only be applied to Euclidean data. However, modern data in the real world are from natural language processing, mathematical formulas, social networks, transportation and sensor networks, computer visions, automations, and biomedical measurements. The Euclidean assumption is not appropriate for non Euclidean data. This perspective addresses the urgent need to overcome those fundamental limitations and encourages extensions of classical probability theory and hypothesis testing , diffusion models and stochastic differential equations from Euclidean space to non Euclidean space. Artificial intelligence such as natural language processing, computer vision, graphical neural networks, manifold regression and inference theory, manifold learning, graph neural networks, compositional diffusion models for automatically compositional generations of concepts and demystifying machine learning systems, has been rapidly developed. Differential manifold theory is the mathematic foundations of deep learning and data science as well. We urgently need to shift the paradigm for data analysis from the classical Euclidean data analysis to both Euclidean and non Euclidean data analysis and develop more and more innovative methods for describing, estimating and inferring non Euclidean geometries of modern real datasets. A general framework for integrated analysis of both Euclidean and non Euclidean data, composite AI, decision intelligence and edge AI provide powerful innovative ideas and strategies for fundamentally advancing AI. We are expected to marry statistics with AI, develop a unified theory of modern statistics and drive next generation of AI and data science.Comment: 37 page