[论文解读] A Newton algorithm for semi-discrete optimal transport

Jun Kitagawa, Quentin Mérigot+1 more|arXiv (Cornell University)|Mar 17, 2016

Point processes and geometric inequalities参考文献 26被引用 18

一句话总结

本文提出了一种用于半离散最优传输的阻尼牛顿算法，结合了实际效率与全局收敛保证。通过利用Ma-Trudinger-Wang条件和加权Poincaré-Wirtinger不等式，该方法实现了最优收敛速率，弥合了收敛缓慢但理论可靠的算法与快速但无保证的求解器之间的差距。

ABSTRACT

Many problems in geometric optics or convex geometry can be recast as optimal transport problems: this includes the far-field reflector problem, Alexandrov's curvature prescription problem, etc. A popular way to solve these problems numerically is to assume that the source probability measure is absolutely continuous while the target measure is finitely supported. We refer to this setting as semi-discrete optimal transport. Among the several algorithms proposed to solve semi-discrete optimal transport problems, one currently needs to choose between algorithms that are slow but come with a convergence speed analysis (e.g. Oliker-Prussner) or algorithms that are much faster in practice but which come with no convergence guarantees Algorithms of the first kind rely on coordinate-wise increments and the number of iterations required to reach the solution up to an error of $\epsilon$ is of order $N^3/\epsilon$, where $N$ is the number of Dirac masses in the target measure. On the other hand, algorithms of the second kind typically rely on the formulation of the semi-discrete optimal transport problem as an unconstrained convex optimization problem which is solved using a Newton or quasi-Newton method. The purpose of this article is to bridge this gap between theory and practice by introducing a damped Newton's algorithm which is experimentally efficient and by proving the global convergence of this algorithm with optimal rates. The main assumptions is that the cost function satisfies a condition that appears in the regularity theory for optimal transport (the Ma-Trudinger-Wang condition) and that the support of the source density is connected in a quantitative way (it must satisfy a weighted Poincar\'e-Wirtinger inequality).

研究动机与目标

• 解决半离散最优传输中理论保证但速度缓慢的算法与快速但无保证的求解器之间的差距。
• 开发一种数值高效的算法，同时保持严格的收敛性分析。
• 在代价函数和源测度支撑的几何与解析条件下，证明全局收敛性并达到最优收敛速率。
• 证明该算法在实现上实用的同时，收敛速率可与理论界相媲美。

提出的方法

• 将半离散最优传输问题表述为无约束凸优化问题。
• 应用阻尼牛顿法求解对偶问题，确保每一步均有充分下降。
• 依赖Ma-Trudinger-Wang条件以确保对偶势函数的强凸性和正则性。
• 在源测度支撑上施加加权Poincaré-Wirtinger不等式，以确保其定量连通性。
• 引入阻尼以稳定牛顿步长，保证全局收敛。
• 该算法旨在避免Oliker-Prussner等方法中缓慢的坐标更新，同时保留理论保证。

实验结果

研究问题

• RQ1 能否为半离散最优传输设计一种牛顿型方法，使其在实践中实现快速收敛，同时具备严格的全局收敛保证？
• RQ2 代价函数和源测度的哪些几何与解析条件足以确保以最优速率收敛？
• RQ3 如何有效将阻尼融入牛顿法中，以在非凸设置下保持全局收敛？
• RQ4 Ma-Trudinger-Wang条件在多大程度上使牛顿法可用于最优传输问题？
• RQ5 对源支撑的定量连通性条件在多大程度上可替代收敛证明中的较弱假设？

主要发现

• 所提出的阻尼牛顿算法在Ma-Trudinger-Wang条件下实现了全局收敛与最优收敛速率。
• 该算法的收敛速率与理论下界一致，迭代次数按O(N^3/ε)缩放，其中误差为ε，与目前已知的最佳理论界相匹配。
• 该方法在实验中表现出高效性，实际性能优于Oliker-Prussner等坐标更新方法，同时保持收敛保证。
• 加权Poincaré-Wirtinger不等式确保了源支撑为收敛所必需的定量连通性。
• 该算法的阻尼牛顿步长可防止发散，并在每次迭代中保证充分下降，从而实现鲁棒性能。