社交网络上闭环推荐系统的结构化正定最优控制综合

中文摘要 AI 辅助

我们研究网络化多主题意见动态的反馈推荐策略设计。推荐的设计被表述为一个无限时域线性二次最优控制问题，该问题偏好与每个智能体当前意见一致的推荐，从而将意见一致性作为参与度的代理。同时，性能指标惩罚极化、偏离意见动态的未受控均衡、偏离当前智能体的意见、推荐努力，并偏好邻居推荐之间的一致性。智能体层面的表示揭示了社交网络的互连结构，并使得结构化控制器和证书综合成为可能。在性能指标的阶段成本严格正定的假设下，仿射最优控制问题分解为一个严格凸的稳态优化（决定最优均衡和仿射控制器偏移）和一个LQR问题（产生静态状态反馈增益）。由此产生的集中式Riccati控制器提供了一个性能基准，而结构化增益则通过基于耗散性和基于$\mathcal H_2$的LMI替代公式获得。可能重叠的证书簇为结构化增益设计提供了可扩展的局部充分条件，而单独的稳态簇分解为稳态优化提供了精确的共识重述。在社交网络上的数值结果说明了闭环行为以及性能与控制器局部性之间的权衡。

英文摘要

We study the design of feedback recommendation policies for networked multi-topic opinion dynamics. The design of recommendations is formulated as an infinite-horizon linear-quadratic optimal control problem that favors suggestions aligned with each agent's current opinion, thereby using opinion alignment as a proxy for engagement. At the same time, the performance index penalizes polarization, deviation from the uncontrolled equilibrium of the opinion dynamics, and from the current agents' opinion, recommendation effort, and favors coherence among neighbors' recommendations. An agent-wise representation exposes the interconnection structure of the social network and enables structured controller and certificate synthesis. Under the assumption that the stage cost of the performance index is strictly positive definite, the affine optimal control problem separates into a strictly convex steady-state optimization, which determines the optimal equilibrium and the affine controller offset, and an LQR problem, which yields the static state-feedback gain. The resulting centralized Riccati controller provides a performance benchmark, while structured gains are obtained from dissipativity-based and $\mathcal H_2$-based LMI surrogate formulations. Possibly overlapping certificate clusters yield scalable local sufficient conditions for the structured gain design, while a separate steady-state cluster decomposition provides an exact consensus reformulation of the steady-state optimization. Numerical results on a social network illustrate the closed-loop behavior and the trade-off between performance and controller locality.

发表机构

• Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, GIPSA-lab（格勒诺布尔阿尔卑斯大学、法国国家科学研究中心、Inria、格勒诺布尔理工学院、GIPSA实验室）
• Univ. Grenoble Alpes, CNRS, Sciences Po Grenoble-UGA, Pacte（格勒诺布尔阿尔卑斯大学、法国国家科学研究中心、格勒诺布尔政治学院、PACTE研究中心）

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