发表机构
HSE University; Trusted AI Research Center RAS; MIRAI; Innopolis University（高等经济大学; 俄罗斯科学院可信人工智能研究中心; MIRAI; 伊诺波利斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对缓慢时变网络，提出加窗切比雪夫方法WAVE，通过重启递推限制网络变化影响，实现与固定网络相同的加速共识与分布式优化收敛速率。
AI 中文摘要
我们研究缓慢时变网络上的平均共识问题及其在分布式优化中的应用。我们提出了WAVE，一种加窗切比雪夫方法，通过有限窗口后重新启动递推来限制网络变化累积效应。若χ界定网络条件数，且相继通信算子满足‖L_{k+1}-L_k‖≤β，则WAVE在O((√χ+min{βχ²,χ})ln(e/ε))轮通信内达到ε-共识。当β=O(χ^{-3/2})时，该速率匹配固定网络的√χ依赖，并在整个网络变化范围内平滑插值。对于分段常数网络，若每次变化发生时被检测到且连续变化间隔至少τ轮，则WAVE在O((√χ+χ/τ)ln(e/ε))轮通信内达到ε-共识。最后，我们将WAVE用作加速分布式优化方法中的共识步骤，该方法针对α-光滑凸局部目标，其平均函数为μ-强凸。对于足够缓慢的网络变化，每个智能体在Õ(√(α/μ)√χ)轮通信内获得全局优化问题的ε-解，匹配固定网络的依赖关系。
英文摘要
We study average consensus over slowly time-varying networks and its application to decentralized optimization. We introduce WAVE, a windowed Chebyshev method that limits the accumulated effect of network variation by restarting the recurrence after finite windows. If $χ$ bounds the network condition number and successive communication operators satisfy $\|L_{k+1}-L_k\|\leqβ$, WAVE reaches $ε$-consensus in $O((\sqrtχ+\min\lbraceβχ^2,χ\rbrace)\ln(e/ε))$ communication rounds. This rate matches the fixed-network $\sqrtχ$ dependence when $β=O(χ^{-3/2})$ and smoothly interpolates across the full range of network variation. For piecewise-constant networks where each change is detected when it occurs and consecutive changes are at least $τ$ rounds apart, WAVE reaches $ε$-consensus in $O((\sqrtχ+χ/τ)\ln(e/ε))$ communication rounds. Finally, we use WAVE as the consensus step in an accelerated decentralized optimization method for $α$-smooth convex local objectives with a $μ$-strongly convex average. For sufficiently slow network variation, every agent obtains an $ε$-solution to the global optimization problem after $\widetilde O(\sqrt{α/μ}\,\sqrtχ)$ communication rounds, matching the fixed-network dependence.
Comments38 pages
