发表机构
Zhejiang University; Nanjing University; Southeast University（浙江大学; 南京大学; 东南大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种新的分布式非光滑非凸随机优化算法，通过在线到非凸转换消除维度依赖，实现更优的样本和通信复杂度。
AI 中文摘要
我们研究了由 $n$ 个节点组成的网络上的分布式非光滑非凸随机优化问题，目标是找到 $(\delta,\epsilon)$-Goldstein 驻点。现有最优算法实现了 $O(\delta^{-1}(\epsilon^{-3}+d\epsilon^{-1}))$ 的样本复杂度和 $\widetilde{O}(\gamma^{-1/2}\delta^{-1}(\epsilon^{-3}+d\epsilon^{-1}))$ 的通信复杂度，其中 $d$ 是问题维度，$\gamma$ 是通信矩阵的谱间隙。然而，对 $d$ 的多项式依赖在高维场景中可能成为主要瓶颈。在本文中，我们提出了一种新算法，实现了 $O(\delta^{-1}\epsilon^{-3})$ 的样本复杂度和 $\widetilde{O}(\gamma^{-1/2}\delta^{-1}\epsilon^{-3})$ 的通信复杂度。主要技术是一种优雅的分布式在线到非凸的转换，将原始问题简化为分布式在线凸优化（D-OCO）问题。我们转换的一个关键性质是，其共识要求可以直接从底层 D-OCO 决策的共识中继承。特别是，这一性质使我们能够建立维度依赖与共识误差之间的显式联系，进而表明对 $d$ 的多项式依赖可以通过仅增加对数级别的额外通信来消除。
英文摘要
We investigate decentralized nonsmooth nonconvex stochastic optimization over a network of $n$ nodes, with the goal of finding an $(δ,ε)$-Goldstein stationary point. The best existing algorithm achieves $O(δ^{-1}(ε^{-3}+dε^{-1}))$ sample complexity and $\widetilde{O}(γ^{-1/2}δ^{-1}(ε^{-3}+dε^{-1}))$ communication complexity, where $d$ is the problem dimension and $γ$ is the spectral gap of the communication matrix. However, the polynomial dependence on $d$ can be a major bottleneck in high-dimensional regimes. In this paper, we propose a novel algorithm that achieves $O(δ^{-1}ε^{-3})$ sample complexity and $\widetilde{O}(γ^{-1/2}δ^{-1}ε^{-3})$ communication complexity. The primary technique is an elegant decentralized online-to-nonconvex conversion that reduces the original problem to a decentralized online convex optimization (D-OCO) problem. A key property of our conversion is that its consensus requirements can be inherited directly from the consensus of the underlying D-OCO decisions. In particular, this property enables us to establish an explicit connection between the dimension dependence and the consensus error, which in turn shows that the polynomial dependence on $d$ can be removed with only logarithmic additional communication.
