提出带噪声的乘法权重更新算法，解决单纯形上的非凸优化问题。
Langevin Multiplicative Weights Update with Applications in Polynomial Portfolio Management
- 在单纯形上引入与几何相关的噪声，改进乘法权重更新。
- 证明算法能非渐近收敛到内部全局最优解。
- 在多项式投资组合管理中验证了高效性。
我们研究定义在单纯形及其乘积上的非凸优化问题。提出一种基于朗之万动力学的乘法权重更新算法（LMWU），通过引入与非欧几里得几何相关的噪声，实现对全局最优解的求解。尽管近年来关于无约束朗之万梯度方法的逃逸鞍点、避免局部极小值及全局收敛性已有理论进展，但带约束条件下的全局优化仍缺乏充分研究。本文证明了LMWU算法可非渐近收敛至内部全局最小值。我们在真实数据集上的多项式投资组合管理任务中验证了该算法的有效性，其中优化一个高度非线性的目标函数至关重要。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。