As a foundation for optimization, convexity is useful beyond the classical settings of Euclidean and Hilbert space. The broader arena of nonpositively curved metric spaces, which includes manifolds like hyperbolic space, as well as metric trees and more general CAT(0) cubical complexes, supports primal tools like proximal operations for geodesically convex functions. However, the lack of linear structure in such spaces complicates dual constructions like subgradients. To address this hurdle, we introduce a new type of subgradient for functions on Hadamard spaces, based on Busemann functions. Our notion supports generalizations of classical stochastic and incremental subgradient methods, with guaranteed complexity bounds. We illustrate with subgradient algorithms for $p$-mean problems in general Hadamard spaces, in particular computing medians in BHV tree space.

翻译：作为优化理论的基础，凸性在欧几里得空间与希尔伯特空间等经典框架之外仍具有重要价值。非正曲率度量空间这一更广阔的领域——包括双曲空间等流形、度量树以及更一般的CAT(0)立方复形——支持对测地凸函数使用邻近算子等基本工具。然而，此类空间线性结构的缺失使得次梯度等对偶构造变得复杂。为克服此障碍，我们基于Busemann函数为Hadamard空间上的函数引入了一种新型次梯度概念。该概念支持经典随机与增量次梯度方法的推广，并具有可保证的复杂度界。我们以一般Hadamard空间中$p$-均值问题的次梯度算法为例进行说明，特别展示了BHV树空间中位数的计算方法。