The Sliced-Wasserstein distance (SW) is a computationally efficient and theoretically grounded alternative to the Wasserstein distance. Yet, the literature on its statistical properties -- or, more accurately, its generalization properties -- with respect to the distribution of slices, beyond the uniform measure, is scarce. To bring new contributions to this line of research, we leverage the PAC-Bayesian theory and a central observation that SW may be interpreted as an average risk, the quantity PAC-Bayesian bounds have been designed to characterize. We provide three types of results: i) PAC-Bayesian generalization bounds that hold on what we refer as adaptive Sliced-Wasserstein distances, i.e. SW defined with respect to arbitrary distributions of slices (among which data-dependent distributions), ii) a principled procedure to learn the distribution of slices that yields maximally discriminative SW, by optimizing our theoretical bounds, and iii) empirical illustrations of our theoretical findings.

The Light-Up puzzle, also known as the AKARI puzzle, has never been solved using modern artificial intelligence (AI) methods. Currently, the most widely used computational technique to autonomously develop solutions involve evolution theory algorithms. This project is an effort to apply new AI techniques for solving the Light-up puzzle faster and more computationally efficient. The algorithms explored for producing optimal solutions include hill climbing, simulated annealing, feed-forward neural network (FNN), and convolutional neural network (CNN). Two algorithms were developed for hill climbing and simulated annealing using 2 actions (add and remove light bulb) versus 3 actions(add, remove, or move light-bulb to a different cell). Both hill climbing and simulated annealing algorithms showed a higher accuracy for the case of 3 actions. The simulated annealing showed to significantly outperform hill climbing, FNN, CNN, and an evolutionary theory algorithm achieving 100% accuracy in 30 unique board configurations. Lastly, while FNN and CNN algorithms showed low accuracies, computational times were significantly faster compared to the remaining algorithms. The GitHub repository for this project can be found at https://github.com/rperera12/AKARI-LightUp-GameSolver-with-DeepNeuralNetworks-and-HillClimb-or-SimulatedAnnealing.