Langevin Multiplicative Weights Update with Applications in Polynomial Portfolio Management

TL;DR

This paper introduces Langevin Multiplicative Weights Update, an algorithm for nonconvex constrained optimization on simplices, with proven convergence to global minima and demonstrated effectiveness in polynomial portfolio management.

Contribution

The paper proposes a novel Langevin-based algorithm for constrained nonconvex optimization with theoretical convergence guarantees and practical validation in portfolio management.

Findings

Proven convergence of LMWU to interior global minima.

Effective optimization of non-linear objectives in portfolio management.

Algorithm outperforms existing methods on real data.

Abstract

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.