The empirical risk minimization (ERM) problem with relative entropy regularization (ERM-RER) is investigated under the assumption that the reference measure is a $\sigma $ -finite measure, and not necessarily a probability measure. Under this assumption, which leads to a generalization of the ERM-RER problem allowing a larger degree of flexibility for incorporating prior knowledge, numerous relevant properties are stated. Among these properties, the solution to this problem, if it exists, is shown to be a unique probability measure, mutually absolutely continuous with the reference measure. Such a solution exhibits a probably-approximately-correct guarantee for the ERM problem independently of whether the latter possesses a solution. For a fixed dataset and under a specific condition, the empirical risk is shown to be a sub-Gaussian random variable when the models are sampled from the solution to the ERM-RER problem. The generalization capabilities of the solution to the ERM-RER problem (the Gibbs algorithm) are studied via the sensitivity of the expected empirical risk to deviations from such a solution towards alternative probability measures. Finally, an interesting connection between sensitivity, generalization error, and lautum information is established.

In this work, we study Stackelberg equilibria for discounted stochastic games. We consider two solution concepts for these games: stationary strong Stackelberg equilibrium (SSSE) and fixed point equilibrium (FPE) solutions. The SSSE solution is obtained by explicitly solving the Stackelberg equilibrium conditions, whereas the FPE can be computed efficiently using value or policy iteration algorithms. However, previous work has overlooked the relationship between these two different solution concepts. Here, we investigate the conditions for existence and equivalence of these solution concepts. Our theoretical results prove that the FPE and SSSE exist and coincide for important classes of games, including myopic follower strategy and team games. This, however, does not hold in general, and we provide numerical examples where one of SSSE or FPE does not exist, or when they both exist, they differ. Our computational results compare the solutions obtained by value iteration, policy iteration, and a mathematical programming formulations for this problem. Finally, we present a discounted stochastic Stackelberg game for a security application to illustrate the solution concepts and the efficiency of the algorithms studied.