社区结构对共识机器学习的影响
Impact of Community Structure on Consensus Machine Learning
摘要：
共识动态支持分散的机器学习，用于分布在云计算集群或物联网上的数据。在这些和其他设置中，人们寻求最小化在一定$ε>0$误差范围内获得共识所需的时间$τ_ε$。$τ_ε$通常取决于底层通信网络的拓扑，并且对于许多算法，$τ_ε$取决于网络的归一化拉普拉斯矩阵的第二小特征值$λ_2\in[0,1]$：$τ_ε\sim\mathcal{O}(λ_2^{-1})$。在这里，我们分析了网络社区结构对$τ_ε$的影响，例如，当计算节点/传感器在空间上聚类时，可能会出现这种情况。我们研究了从随机块模型中提取的网络上的共识机器学习，该模型产生的随机网络可以包含具有不同大小和密度的异构社区。使用随机矩阵理论，我们分析了社区对$λ_2$和共识的影响，发现$λ_2$通常会增加（即，$τ_ε$减小）。我们进一步观察到，社区结构存在一个临界水平，在这个临界水平上，$τ_ε$达到一个下界，不再受社区存在的限制。我们支持我们的研究结果与分散的支持向量机的实证实验。
英文摘要：
Consensus dynamics support decentralized machine learning for data that is distributed across a cloud compute cluster or across the internet of things. In these and other settings, one seeks to minimize the time $τ_ε$ required to obtain consensus within some $ε>0$ margin of error. $τ_ε$ typically depends on the topology of the underlying communication network, and for many algorithms $τ_ε$ depends on the second-smallest eigenvalue $λ_2\in[0,1]$ of the network's normalized Laplacian matrix: $τ_ε\sim\mathcal{O}(λ_2^{-1})$. Here, we analyze the effect on $τ_ε$ of network community structure, which can arise when compute nodes/sensors are spatially clustered, for example. We study consensus machine learning over networks drawn from stochastic block models, which yield random networks that can contain heterogeneous communities with different sizes and densities. Using random matrix theory, we analyze the effects of communities on $λ_2$ and consensus, finding that $λ_2$ generally increases (i.e., $τ_ε$ decreases) as one decreases the extent of community structure. We further observe that there exists a critical level of community structure at which $τ_ε$ reaches a lower bound and is no longer limited by the presence of communities. We support our findings with empirical experiments for decentralized support vector machines.
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