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Langevin Multiplicative Weights Update with Applications in Polynomial Portfolio Management

Yi Feng, Xiao Wang, Tian Xie

TL;DR

The paper introduces Langevin Multiplicative Weights Update (LMWU), a noise-augmented MWU that respects Shahshahani geometry to solve constrained nonconvex optimization on a product of simplices. By embedding Langevin noise into MWU on the simplex and leveraging Riemannian gradient steps with appropriate retractions, the authors establish non-asymptotic convergence to interior global minima under manifold-specific assumptions. They prove a finite-sample bound showing how the expected objective approaches the global optimum as a function of iteration, noise scale, and temperature, and demonstrate improved performance on polynomial portfolio optimization with real NASDAQ data compared to standard MWU and related methods. The work integrates geometric Langevin dynamics with multi-agent optimization, offering a principled approach to global optimization under simplex constraints with practical impact in portfolio design and other constrained nonconvex problems.

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.

Langevin Multiplicative Weights Update with Applications in Polynomial Portfolio Management

TL;DR

The paper introduces Langevin Multiplicative Weights Update (LMWU), a noise-augmented MWU that respects Shahshahani geometry to solve constrained nonconvex optimization on a product of simplices. By embedding Langevin noise into MWU on the simplex and leveraging Riemannian gradient steps with appropriate retractions, the authors establish non-asymptotic convergence to interior global minima under manifold-specific assumptions. They prove a finite-sample bound showing how the expected objective approaches the global optimum as a function of iteration, noise scale, and temperature, and demonstrate improved performance on polynomial portfolio optimization with real NASDAQ data compared to standard MWU and related methods. The work integrates geometric Langevin dynamics with multi-agent optimization, offering a principled approach to global optimization under simplex constraints with practical impact in portfolio design and other constrained nonconvex problems.

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.

Paper Structure

This paper contains 34 sections, 8 theorems, 125 equations, 9 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Other related works.
  3. Preliminaries
  4. Riemannian Geometry
  5. Riemannian metric and exponential map.
  6. Riemannian gradient.
  7. Retraction.
  8. Distributions on manifold
  9. KL divergence.
  10. Wasserstein distance.
  11. Log-Sobolev inequality.
  12. Main Results
  13. From MWU to Langevin MWU
  14. Main Theorem
  15. Outline of Proof
  16. ...and 19 more sections

Key Result

Theorem 1.1

Suppose the global optima of Problem setup is in the interior of the constraints. The Langevin Multiplicative Weights Update converges to the biased global optima in expectation.

Figures (9)

  • Figure 1: Comparison of LMWU with Accelerated MWU FPW2022, Projected Lagevin Lamperski, and PRGD CB19.
  • Figure 2: Future comparison of LMWU with Projected Langevin Lamperski.
  • Figure 3: An Illustration of Non-convexity in Polynomial Portfolio Optimization
  • Figure 4: A 3D Illustration of Non-convexity in Polynomial Portfolio Optimization
  • Figure 5: Test function $f_1$
  • ...and 4 more figures

Theorems & Definitions (13)

  • Theorem 1.1: Informal
  • Theorem 3.1
  • Lemma 3.1
  • Lemma 3.2
  • Theorem E.1: Fokker-Planck Equation
  • Proposition E.2
  • proof
  • proof
  • proof
  • Lemma G.1
  • ...and 3 more