We consider nonconvex optimization problem over simplex, and more generally, a product of simplices. We provide an algorithm, Langevin Multiplicative Weights Update (LMWU) for solving global optimization problems by adding a noise scaling with the non-Euclidean geometry in the simplex. Non-convex optimization has been extensively studied by machine learning community due to its application in various scenarios such as neural network approximation and finding Nash equilibrium. Despite recent progresses on provable guarantee of escaping and avoiding saddle point (convergence to local minima) and global convergence of Langevin gradient based method without constraints, the global optimization with constraints is less studied. We show that LMWU algorithm is provably convergent to interior global minima with a non-asymptotic convergence analysis. We verify the efficiency of the proposed algorithm in real data set from polynomial portfolio management, where optimization of a highly non-linear objective function plays a crucial role.

In view of the complexity of the dynamics of learning in games, we seek to decompose a game into simpler components where the dynamics' long-run behavior is well understood. A natural starting point for this is Helmholtz's theorem, which decomposes a vector field into a potential and an incompressible component. However, the geometry of game dynamics - and, in particular, the dynamics of exponential / multiplicative weights (EW) schemes - is not compatible with the Euclidean underpinnings of Helmholtz's theorem. This leads us to consider a specific Riemannian framework based on the so-called Shahshahani metric, and introduce the class of incompressible games, for which we establish the following results: First, in addition to being volume-preserving, the continuous-time EW dynamics in incompressible games admit a constant of motion and are Poincar\'e recurrent - i.e., almost every trajectory of play comes arbitrarily close to its starting point infinitely often. Second, we establish a deep connection with a well-known decomposition of games into a potential and harmonic component (where the players' objectives are aligned and anti-aligned respectively): a game is incompressible if and only if it is harmonic, implying in turn that the EW dynamics lead to Poincar\'e recurrence in harmonic games.