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 $\mathcal{O}(n^2\sqrt{1/\varepsilon}\ln(1/δ))$.

翻译：非平衡最优传输（UOT）已成为标准最优传输的一种鲁棒松弛形式，在处理异常值和质量变化方面尤为有效。然而，针对UOT的可扩展算法，特别是基于随机梯度下降（SGD）的方法，目前仍缺乏深入探索。本研究通过分析熵正则化UOT的半对偶形式，证明了其适用于自适应梯度方法，从而填补了这一空白。虽然半对偶是大规模平衡OT的标准工具，但在非平衡设定下，其几何性质在标准分析中表现出病态特性。具体而言，使用χ²散度的边际惩罚最坏情况边界表明条件数与n/ε成比例，这意味着可扩展性较差。与之相反，我们证明局部条件数实际按O(1/ε)缩放，有效消除了对n的病态依赖。利用这一特性，我们证明了SGD方法能够适应这种局部曲率，在随机和在线机制下达到O(n/εT)的收敛速率，使其适用于大规模和半离散应用场景。最后，针对全批量离散设定，我们推导出仅依赖于梯度的局部平滑度的近乎紧致上界。通过利用该上界自适应调整步长，我们在半对偶泛函上提出改进的自适应涅斯捷罗夫加速梯度（ANAG）方法，并证明其达到O(n²√(1/ε)ln(1/δ))的局部计算复杂度。