智能体网络信息聚合的最优速率
Optimal Rates for Agentic Networked Information Aggregation
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- Google Research（谷歌研究院）
- University of California, Irvine（加州大学欧文分校）
- University of Maryland（马里兰大学）
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中文总结 AI 辅助
本文针对网络学习模型中智能体信息聚合的额外均方误差速率，填补了已有研究的下界差距，证明了线性回归和逻辑分类任务的最优速率，并给出了相应的实例与分析。
中文摘要 AI 辅助
本文基于Kearns、Roth和Ryu在SODA'26上的开创性论文，研究网络学习模型中的信息聚合问题。该模型捕捉了智能体AI的核心模式：每个智能体仅能看到部分数据，且仅传递自身的结论。该模型考虑了带有均方误差（MSE）损失的线性回归问题，智能体位于有向无环图（DAG）中，每个智能体仅能看到部分特征及其父节点的预测结果，拟合线性预测器后仅将自身预测结果向前传递，基准为能看到所有原始特征的全特征学习器。若一条深度为$D$的路径满足每$M$个连续智能体组成的块能共同看到所有原始特征，则称该路径为$M$-覆盖路径。Kearns等人证明此类路径最后一个智能体的额外均方误差为$O(M/\sqrt{D})$，并给出了在$D<M^2$时额外误差为$\Omega(M/D)$的循环实例。本文填补了这一差距：在深度不超过$M^2$时，额外误差为常数级；在深度超过$M^2$时，额外误差为$\Theta(M^2/D)$。本文首先对循环实例进行更精确的分析，将其下界改进为$D<M^2$时的$\Omega(\sqrt{M/D})$；接着对每个$D\ge M^2$，构造了一条深度为$D$的$M$-覆盖路径，其额外误差为$\Omega(M^2/D)$，该实例还给出了所有$D<M^2$时的常数下界。本文还证明，对于任意固定分布，额外误差会沿路径几何级数收缩，排除了在每个深度都存在多项式下界的单一实例。最后，本文在Bateni等人的logit传递模型中对逻辑分类问题证明了相同的最优速率，该模型考虑二元交叉熵（BCE）损失，同样适用$O(M^2/D)$的改进上界，本文还通过证明在这些实例中逻辑路径按比例缩放后与最小二乘路径一致，将所有回归下界进行了迁移。
英文摘要
Building on the pioneering paper of Kearns, Roth, and Ryu (SODA'26), we study information aggregation in a networked learning model. The model captures a central pattern in agentic AI: each agent sees only part of the data and passes on only its own conclusion. Their model considers a linear regression problem with the mean squared error (MSE) loss. Agents sit in a DAG and each sees only a subset of the features and its parents' predictions, fits a linear predictor, and passes only its prediction forward. The benchmark is the full-feature learner that sees all raw features. A path of depth $D$ is $M$-covered if every block of $M$ consecutive agents collectively sees all raw features. Kearns, Roth, and Ryu proved that the excess mean squared error of the last agent on such a path is $O(M/\sqrt D)$, and gave a cyclic instance with excess error $Ω(M/D)$ for $D<M^2$. We close this gap: the correct rate is constant up to depth $M^2$, and $Θ(M^2/D)$ beyond it. We first give a sharper analysis of the cyclic instance and improve its lower bound to $Ω(\sqrt{M/D})$ for $D<M^2$. We then construct, for every depth $D\ge M^2$, an $M$-covered path of depth $D$ with excess error $Ω(M^2/D)$. The same instance gives the constant lower bound for all $D < M^2$. We also show that for any fixed distribution the excess error contracts geometrically along the path, ruling out any single instance that witnesses any polynomial lower bound at every depth. Finally, we prove the same optimal rate for logistic classification in the logit-passing model of Bateni et al., which considers the binary cross-entropy (BCE) loss. The same improved upper bound of $O(M^2/D)$ holds, and we transfer all the regression lower bounds by showing that on those examples the logistic path follows the least-squares path up to rescaling.
