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arXiv 2602.10697math.OCstat.ML
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快速且大规模的不平衡最优传输 via 其半对偶和自适应梯度方法

Fast and Large-Scale Unbalanced Optimal Transport via its Semi-Dual and Adaptive Gradient Methods

Sorbonne Université（索邦大学）

Ferdinand Genans

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AI总结：

本文提出了一种基于半对偶和自适应梯度方法的快速大规模不平衡最优传输算法，通过分析局部曲率特性，实现了高效的收敛率和可扩展性。

摘要：

不平衡最优传输（UOT）作为标准最优传输的稳健放松，尤其有效于处理异常值和质量变化。然而，基于梯度下降（SGD）的可扩展算法仍鲜有研究。本文通过分析熵UOT的半对偶形式，展示了其适用于自适应梯度方法的适宜性。虽然半对偶是大规模平衡OT的标准工具，但在不平衡设定中，其几何结构在标准分析下显得病态。具体而言，使用χ²散度对边际惩罚的最坏界表明，条件数与n/ε成比例，暗示了较差的可扩展性。相反，我们证明局部条件数实际上与1/ε成比例，有效去除了对n的病态依赖。利用这一性质，我们证明SGD方法能够适应这种局部曲率，实现随机和在线情形下的收敛率O(n/εT)，使其适用于大规模和半离散应用。最后，在全批量离散设置中，我们推导了一个几乎紧的局部平滑度上界，仅依赖于梯度。利用它来调整步长，我们提出了一种改进的自适应Nesterov加速梯度（ANAG）方法，并证明其在半对偶功能上实现了局部复杂度O(n²√(1/ε)ln(1/δ))。

英文摘要：

Unbalanced Optimal Transport (UOT) has emerged as a robust relaxation of standard Optimal Transport, particularly effective for handling outliers and mass variations. However, scalable algorithms for UOT, specifically those based on Gradient Descent (SGD), remain largely underexplored. In this work, we address this gap by analyzing the semi-dual formulation of Entropic UOT and demonstrating its suitability for adaptive gradient methods. While the semi-dual is a standard tool for large-scale balanced OT, its geometry in the unbalanced setting appears ill-conditioned under standard analysis. Specifically, worst-case bounds on the marginal penalties using $χ^2$ divergence suggest a condition number scaling with $n/\varepsilon$, implying poor scalability. In contrast, we show that the local condition number actually scales as $\mathcal{O}(1/\varepsilon)$, effectively removing the ill-conditioned dependence on $n$. Exploiting this property, we prove that SGD methods adapt to this local curvature, achieving a convergence rate of $\mathcal{O}(n/\varepsilon T)$ in the stochastic and online regimes, making it suitable for large-scale and semi-discrete applications. Finally, for the full batch discrete setting, we derive a nearly tight upper bound on local smoothness depending solely on the gradient. Using it to adapt step sizes, we propose a modified Adaptive Nesterov Accelerated Gradient (ANAG) method on the semi-dual functional and prove that it achieves a local complexity of $\mathcal{O}(n^2\sqrt{1/\varepsilon}\ln(1/δ))$.

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