We consider the Online Facility Location (OFL) problem in the framework of learning-augmented online algorithms. In Online Facility Location (OFL), demands arrive one-by-one in a metric space and must be (irrevocably) assigned to an open facility upon arrival, without any knowledge about future demands. We focus on uniform facility opening costs and present an online algorithm for OFL that exploits potentially imperfect predictions on the locations of the optimal facilities. We prove that the competitive ratio decreases from sublogarithmic in the number n of demands to constant as the so-called η1 error, i.e., the sum of distances of the predicted locations to the optimal facility locations, decreases towards zero. E.g., our analysis implies that if for some ε > 0, η1 = OPT / n^ε, where OPT is the cost of the optimal solution, the competitive ratio is O(1/ε). We complement our analysis with a matching lower bound establishing that the dependence of the algorithm's competitive ratio on the η1 error is optimal, up to constant factors.

We consider the problem of computing Nash equilibria in potential games where each player's strategy set is subject to private uncoupled constraints. This scenario is frequently encountered in real-world applications like road network congestion games where individual drivers adhere to personal budget and fuel limitations. Despite the plethora of algorithms that efficiently compute Nash equilibria (NE) in potential games, the domain of constrained potential games remains largely unexplored. We introduce an algorithm that leverages the Lagrangian formulation of NE. The algorithm is implemented independently by each player and runs in polynomial time with respect to the approximation error, the sum of the size of the action-spaces, and the game's inherit parameters.

Learning Nash equilibria (NE) in games has garnered significant attention, particularly in the context of training Generative Adversarial Networks (GANs) and multi-agent Reinforcement Learning. The current state-of-the-art in efficiently learning games focuses on landscapes that meet the (weak) Minty property or games characterized by a unique function, often referred to as potential games. A significant challenge in this domain is that computing Nash equilibria is a computationally intractable task [Daskalakis et al. 2009]. In this paper we focus on bimatrix games (A,B) called rank-1. These are games in which the sum of the payoff matrices A+B is a rank 1 matrix; note that standard zero-sum games are rank 0. We show that optimistic gradient descent/ascent converges to an \epsilon-approximate NE after 1/\epsilon^2 log(1/\epsilon) iterates in rank-1 games. We achieve this by leveraging structural results about the NE landscape of rank-1 games Adsul et al. 2021. Notably, our approach bypasses the fact that these games do not satisfy the MVI property.

Fictitious Play (FP) is a simple and natural dynamic for repeated play with many applications in game theory and multi-agent reinforcement learning. It was introduced by Brown and its convergence properties for two-player zero-sum games was established later by Robinson. Potential games [Monderer and Shapley 1996] is another class of games which exhibit the FP property [Monderer and Shapley 1996], i.e., FP dynamics converges to a Nash equilibrium if all agents follows it. Nevertheless, except for two-player zero-sum games and for specific instances of payoff matrices [Abernethy et. al. 2021] or for adversarial tie-breaking rules [Daskalakis and Pan, 2014], the \textit{convergence rate} of FP is unknown. In this work, we focus on the rate of convergence of FP when applied to potential games and more specifically identical payoff games. We prove that FP can take exponential time (in the number of strategies) to reach a Nash equilibrium, even if the game is restricted to \textit{two agents}. To prove this, we recursively construct a two-player coordination game with a unique Nash equilibrium. Moreover, every approximate Nash equilibrium in the constructed game must be close to the pure Nash equilibrium in $\ell_1$-distance.