Consider public health officials aiming to spread awareness about a new vaccine in a community interconnected by a social network. How can they distribute information with minimal resources, so as to avoid polarization and ensure community-wide convergence of opinion? To tackle such challenges, we initiate the study of sample complexity of opinion convergence in networks. Our framework is built on the recognized opinion formation game, where we regard the opinion of each agent as a data-derived model, unlike previous works that treat opinions as data-independent scalars. The opinion model for every agent is initially learned from its local samples and evolves game-theoretically as all agents communicate with neighbors and revise their models towards an equilibrium. Our focus is on the sample complexity needed to ensure that the opinions converge to an equilibrium such that the final model of every agent has low generalization error. Our paper has two main technical results. First, we present a novel polynomial time optimization framework to quantify the total sample complexity for arbitrary networks, when the underlying learning problem is (generalized) linear regression. Second, we leverage this optimization to study the network gain which measures the improvement of sample complexity when learning over a network compared to that in isolation. Towards this end, we derive network gain bounds for various network classes including cliques, star graphs, and random regular graphs. Additionally, our framework provides a method to study sample distribution within the network, suggesting that it is sufficient to allocate samples inversely to the degree. Empirical results on both synthetic and real-world networks strongly support our theoretical findings.

翻译：考虑公共卫生官员试图在通过社交网络相互连接的社区中传播关于新疫苗的认知。他们如何能以最少的资源分配信息，从而避免观点极化并确保社区范围内的观点收敛？为应对此类挑战，我们开创性地研究了网络中观点收敛的样本复杂度问题。我们的框架建立在公认的观点形成博弈之上，其中我们将每个智能体的观点视为数据驱动的模型，这与先前将观点视为数据无关标量的研究不同。每个智能体的观点模型最初从其本地样本中学习获得，并随着所有智能体与邻居通信并修正其模型以趋向均衡而进行博弈演化。我们的研究重点在于确保观点收敛至均衡状态所需的样本复杂度，使得每个智能体的最终模型具有较低的泛化误差。本文包含两项主要技术成果：首先，当底层学习问题为（广义）线性回归时，我们提出了一种新颖的多项式时间优化框架来量化任意网络的总样本复杂度；其次，我们利用该优化方法研究网络增益——即衡量在网络中学习相较于孤立学习时样本复杂度的改进程度。为此，我们推导了多种网络类型（包括团图、星形图和随机正则图）的网络增益边界。此外，我们的框架提供了一种研究网络内样本分布的方法，表明按节点度数的倒数分配样本即可满足需求。在合成网络和真实网络上的实证结果有力支持了我们的理论发现。