Langevin Multiplicative Weights Update with Applications in Polynomial Portfolio Management
It addresses global optimization with constraints, a less studied problem in machine learning, with applications in portfolio management, though it appears incremental as it builds on existing Langevin and multiplicative weights methods.
The paper tackles nonconvex optimization over simplices by proposing the Langevin Multiplicative Weights Update (LMWU) algorithm, which provably converges to interior global minima with non-asymptotic analysis and demonstrates efficiency on real polynomial portfolio management data.
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.