We study zero-sum games in the space of probability distributions over the Euclidean space $\mathbb{R}^d$ with entropy regularization, in the setting when the interaction function between the players is smooth and strongly convex-concave. We prove an exponential convergence guarantee for the mean-field min-max Langevin dynamics to compute the equilibrium distribution of the zero-sum game. We also study the finite-particle approximation of the mean-field min-max Langevin dynamics, both in continuous and discrete times. We prove biased convergence guarantees for the continuous-time finite-particle min-max Langevin dynamics to the stationary mean-field equilibrium distribution with an explicit bias estimate which does not scale with the number of particles. We also prove biased convergence guarantees for the discrete-time finite-particle min-max Langevin algorithm to the stationary mean-field equilibrium distribution with an additional bias term which scales with the step size and the number of particles. This provides an explicit iteration complexity for the average particle along the finite-particle algorithm to approximately compute the equilibrium distribution of the zero-sum game.

Motivated by understanding the behavior of the Alternating Mirror Descent (AMD) algorithm for bilinear zero-sum games, we study the discretization of continuous-time Hamiltonian flow via the symplectic Euler method. We provide a framework for analysis using results from Hamiltonian dynamics, Lie algebra, and symplectic numerical integrators, with an emphasis on the existence and properties of a conserved quantity, the modified Hamiltonian (MH), for the symplectic Euler method. We compute the MH in closed-form when the original Hamiltonian is a quadratic function, and show that it generally differs from the other conserved quantity known previously in that case. We derive new error bounds on the MH when truncated at orders in the stepsize in terms of the number of iterations, $K$, and use these bounds to show an improved $\mathcal{O}(K^{1/5})$ total regret bound and an $\mathcal{O}(K^{-4/5})$ duality gap of the average iterates for AMD. Finally, we propose a conjecture which, if true, would imply that the total regret for AMD scales as $\mathcal{O}\left(K^{\varepsilon}\right)$ and the duality gap of the average iterates as $\mathcal{O}\left(K^{-1+\varepsilon}\right)$ for any $\varepsilon>0$, and we can take $\varepsilon=0$ upon certain convergence conditions for the MH.

Mild Traumatic Brain Injury (mTBI) is a significant public health problem. The most troubling symptoms after mTBI are cognitive complaints. Studies show measurable differences between patients with mTBI and healthy controls with respect to tissue microstructure using diffusion MRI. However, it remains unclear which diffusion measures are the most informative with regard to cognitive functions in both the healthy state as well as after injury. In this study, we use diffusion MRI to formulate a predictive model for performance on working memory based on the most relevant MRI features. The key challenge is to identify relevant features over a large feature space with high accuracy in an efficient manner. To tackle this challenge, we propose a novel improvement of the best first search approach with crossover operators inspired by genetic algorithm. Compared against other heuristic feature selection algorithms, the proposed method achieves significantly more accurate predictions and yields clinically interpretable selected features.