Existing convergence of distributed optimization methods in non-Euclidean geometries typically rely on kernel assumptions: (i) global Lipschitz smoothness and (ii) bi-convexity of the associated Bregman divergence function. Unfortunately, these conditions are violated by nearly all kernels used in practice, leaving a huge theory-practice gap. This work closes this gap by developing a unified analytical tool that guarantees convergence under mild conditions. Specifically, we introduce Hessian relative uniform continuity (HRUC), a regularity satisfied by nearly all standard kernels. Importantly, HRUC is closed under concatenation, positive scaling, composition, and various kernel combinations. Leveraging the geometric structure induced by HRUC, we derive convergence guarantees for mirror descent-based gradient tracking without imposing any restrictive assumptions. More broadly, our analysis techniques extend seamlessly to other decentralized optimization methods in genuinely non-Euclidean and non-Lipschitz settings.

翻译：现有非欧几何中分布式优化方法的收敛性分析通常依赖于核假设：(i) 全局Lipschitz光滑性及(ii)相关Bregman散度函数的双凸性。遗憾的是，实际使用的几乎所有核函数均不满足这些条件，导致理论与实践的严重脱节。本研究通过构建在温和条件下保证收敛的统一分析工具来弥合这一鸿沟。具体而言，我们提出Hessian相对一致连续性（HRUC）这一正则条件，几乎所有标准核函数均满足该条件。值得注意的是，HRUC在串联、正缩放、复合及多种核组合操作下具有封闭性。借助HRUC诱导的几何结构，我们在不施加任何限制性假设的前提下，为基于镜像下降的梯度跟踪方法建立了收敛性保证。更广泛而言，我们的分析技术可无缝扩展到真正非欧与非Lipschitz场景下的其他去中心化优化方法。