mlsanityData to 14 Oct 2026

Langevin Multiplicative Weights Update with Applications in Polynomial Portfolio Management

arXiv 2502.19210 ↗First posted 26 Feb 2025math.OC cs.LGcited by 0

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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.

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.

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References · 29

  1. 2017How to Escape Saddle Points Efficientlycited by 285
  2. 2017Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysiscited by 243
  3. 2019Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Sufficescited by 227
  4. 2014Theoretical guarantees for approximate sampling from smooth and log-concave densitiescited by 208
  5. 2015Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithmcited by 197
  6. 2018Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problemcited by 123
  7. 2017Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimizationcited by 97
  8. 2017Accelerated Gradient Descent Escapes Saddle Points Faster than Gradient Descentcited by 66
  9. 2018Mirrored Langevin Dynamicscited by 61
  10. 2020Wasserstein Control of Mirror Langevin Monte Carlocited by 55
  11. 2021The Mirror Langevin Algorithm Converges with Vanishing Biascited by 37
  12. 2019First-order methods almost always avoid saddle points: the case of vanishing step-sizescited by 35
17 more
  1. 2016Gradient Descent Only Converges to Minimizers: Non-Isolated Critical Points and Invariant Regionscited by 34
  2. 2019Escaping from saddle points on Riemannian manifoldscited by 33
  3. 2020Projected Stochastic Gradient Langevin Algorithms for Constrained Sampling and Non-Convex Learningcited by 33
  4. 2017Multiplicative Weights Update with Constant Step-Size in Congestion Games: Convergence, Limit Cycles and Chaoscited by 29
  5. 2019Vortices Instead of Equilibria in MinMax Optimization: Chaos and Butterfly Effects of Online Learning in Zero-Sum Gamescited by 25
  6. 2022Convergence of the Riemannian Langevin Algorithmcited by 25
  7. 2019Efficiently escaping saddle points on manifoldscited by 24
  8. 2020Riemannian Langevin Algorithm for Solving Semidefinite Programscited by 22
  9. 2020Chaos, Extremism and Optimism: Volume Analysis of Learning in Gamescited by 16
  10. 2020Fast Convergence of Langevin Dynamics on Manifold: Geodesics meet Log-Sobolevcited by 16
  11. 2022Improved Convergence Rate of Stochastic Gradient Langevin Dynamics with Variance Reduction and its Application to Optimizationcited by 16
  12. 2019Fast and Furious Learning in Zero-Sum Games: Vanishing Regret with Non-Vanishing Step Sizescited by 14
  13. 2019Heavy-ball Algorithms Always Escape Saddle Pointscited by 13
  14. 2023On the Last-iterate Convergence in Time-varying Zero-sum Games: Extra Gradient Succeeds where Optimism Failscited by 8
  15. 2018Multiplicative Weights Update as a Distributed Constrained Optimization Algorithm: Convergence to Second-order Stationary Points Almost Alwayscited by 7
  16. 2024Riemannian Accelerated Zeroth-order Algorithm: Improved Robustness and Lower Query Complexitycited by 5
  17. 2024Prediction Accuracy of Learning in Games : Follow-the-Regularized-Leader meets Heisenbergcited by 2

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