# Langevin Multiplicative Weights Update with Applications in Polynomial Portfolio Management

arXiv: https://arxiv.org/abs/2502.19210 · HTML: https://mlsanity.com/paper/2502.19210/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 29 of its 48 references that are in the map.
- First posted on arXiv 26 Feb 2025.

## Builds on (most cited first, 29 of 29)

- [How to Escape Saddle Points Efficiently](https://mlsanity.com/paper/1703.00887/) (2017)
- [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimization](https://mlsanity.com/paper/1707.06618/) (2017)
- [Accelerated Gradient Descent Escapes Saddle Points Faster than Gradient Descent](https://mlsanity.com/paper/1711.10456/) (2017)
- [Mirrored Langevin Dynamics](https://mlsanity.com/paper/1802.10174/) (2018)
- [Wasserstein Control of Mirror Langevin Monte Carlo](https://mlsanity.com/paper/2002.04363/) (2020)
- [The Mirror Langevin Algorithm Converges with Vanishing Bias](https://mlsanity.com/paper/2109.12077/) (2021)
- [First-order methods almost always avoid saddle points: the case of vanishing step-sizes](https://mlsanity.com/paper/1906.07772/) (2019)
- [Gradient Descent Only Converges to Minimizers: Non-Isolated Critical Points and Invariant Regions](https://mlsanity.com/paper/1605.00405/) (2016)
- [Escaping from saddle points on Riemannian manifolds](https://mlsanity.com/paper/1906.07355/) (2019)
- [Projected Stochastic Gradient Langevin Algorithms for Constrained Sampling and Non-Convex Learning](https://mlsanity.com/paper/2012.12137/) (2020)
- [Multiplicative Weights Update with Constant Step-Size in Congestion Games: Convergence, Limit Cycles and Chaos](https://mlsanity.com/paper/1703.01138/) (2017)
- [Vortices Instead of Equilibria in MinMax Optimization: Chaos and Butterfly Effects of Online Learning in Zero-Sum Games](https://mlsanity.com/paper/1905.08396/) (2019)
- [Convergence of the Riemannian Langevin Algorithm](https://mlsanity.com/paper/2204.10818/) (2022)
- [Efficiently escaping saddle points on manifolds](https://mlsanity.com/paper/1906.04321/) (2019)
- [Riemannian Langevin Algorithm for Solving Semidefinite Programs](https://mlsanity.com/paper/2010.11176/) (2020)
- [Chaos, Extremism and Optimism: Volume Analysis of Learning in Games](https://mlsanity.com/paper/2005.13996/) (2020)
- [Fast Convergence of Langevin Dynamics on Manifold: Geodesics meet Log-Sobolev](https://mlsanity.com/paper/2010.05263/) (2020)
- [Improved Convergence Rate of Stochastic Gradient Langevin Dynamics with Variance Reduction and its Application to Optimization](https://mlsanity.com/paper/2203.16217/) (2022)
- [Fast and Furious Learning in Zero-Sum Games: Vanishing Regret with Non-Vanishing Step Sizes](https://mlsanity.com/paper/1905.04532/) (2019)
- [Heavy-ball Algorithms Always Escape Saddle Points](https://mlsanity.com/paper/1907.09697/) (2019)
- [On the Last-iterate Convergence in Time-varying Zero-sum Games: Extra Gradient Succeeds where Optimism Fails](https://mlsanity.com/paper/2310.02604/) (2023)
- [Multiplicative Weights Update as a Distributed Constrained Optimization Algorithm: Convergence to Second-order Stationary Points Almost Always](https://mlsanity.com/paper/1810.05355/) (2018)
- [Riemannian Accelerated Zeroth-order Algorithm: Improved Robustness and Lower Query Complexity](https://mlsanity.com/paper/2405.05713/) (2024)
- [Prediction Accuracy of Learning in Games : Follow-the-Regularized-Leader meets Heisenberg](https://mlsanity.com/paper/2406.10603/) (2024)

## Abstract (arXiv)

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.