AI 中文总结
针对带异质性曲率的标量二次问题的分布式梯度跟踪，提出一种可证收敛的自适应步长规则，其性能优于现有步长选择规则。
AI 中文摘要
在本研究中，我们针对应用于具有异质性曲率的标量二次问题的分布式梯度跟踪算法，提出了一种具有可证收敛性的自适应步长规则。大多数基于分布式梯度的算法都需要合适的步长选择，现有理论界的步长界往往过于保守，而实际实现通常依赖经验调整的启发式方法。在线自适应策略仅在近期才在一般分布式凸优化中出现，但其性质和性能仍仅被部分理解。为获得分析性见解，我们聚焦于具有信息价值的标量二次成本场景，该场景可明确捕捉网络拓扑与曲率异质性之间的相互作用。我们推导了一个收敛界，该界仅由共识矩阵的本质谱半径和局部成本曲率的异质性参数化，两者均可在线计算，无需优化问题的任何先验知识。优化该界可得到收敛率的计算易处理的替代形式及最优常数步长。所得步长具有解析解释，可保证任意网络拓扑和曲率异质性下的收敛性，且对于完全图和同质曲率是可证紧的。最后，大量数值模拟表明，所提出的分布式自适应策略在考虑的场景中显著优于现有离线和在线步长选择规则。
英文摘要
In this work, we propose an adaptive stepsize rule with guaranteed convergence for Distributed Gradient Tracking applied to scalar quadratic problems with heterogeneous curvatures. Most distributed gradient-based algorithms require a suitable stepsize selection. Available theoretical bounds are often overly conservative, while practical implementations typically rely on empirically tuned heuristics. Online adaptive strategies have only recently emerged for general distributed convex optimization, but their properties and performance remain only partially understood. To gain analytical insight, we focus on the informative setting of scalar quadratic costs, which allows us to explicitly capture the interplay between network topology and curvature heterogeneity. We derive a convergence bound parameterized only by the essential spectral radius of the consensus matrix and the heterogeneity of the local cost curvatures, both computable online without any prior knowledge of the optimization problem. Optimizing this bound yields a computationally tractable surrogate for the convergence rate and the optimal constant stepsize. The resulting stepsize admits an analytical interpretation, guarantees convergence for arbitrary network topologies and curvature heterogeneity, and is provably tight for complete graphs and homogeneous curvatures. Finally, extensive numerical simulations demonstrate that the proposed distributed adaptive strategy significantly outperforms existing offline and online stepsize selection rules in the considered setting.
