Optimal Networks for Agentic Information Aggregation

We study information aggregation in the networked learning model introduced by Kearns et al. (SODA 2026). There is a fixed distribution over $d$ features and a common label. Agents learn in topological order on a directed acyclic graph. Each observes a subset of the features and its parents' predictions, fits a linear predictor to minimize mean squared error, and passes only its prediction forward. The global predictor is the best linear predictor using all features. Kearns et al. show that the output agent's error approaches the global predictor's error along sufficiently deep paths with suitable feature coverage, while insufficient depth can prevent aggregation even in large networks. In contrast to their main focus on a given graph and feature allocation, we consider the limits of the model under two settings. In the *adaptive designer* setting, a designer chooses the graph, feature allocation, and output agent knowing the distribution. In the *oblivious designer* setting, the designer fixes all three before an adversary chooses the distribution. Each agent observes one feature and receives predictions from a limited number of parents. We call the aggregation *exact* when the output agent matches the global predictor exactly. For $d\ge 3$, we show that no finite depth guarantees exact aggregation for every distribution with one parent per agent, even when the designer knows the distribution. In contrast, two parents per agent suffice for exact aggregation even in the oblivious designer setting. A fixed graph, feature allocation, and output agent achieve this for every distribution at depth $O(d\log d)$. Knowing the distribution reduces the depth to $O(d)$. Both constructions use $O(d^2)$ agents, with a very large constant for two parents. We show the bounds on the depth and number of agents are all optimal up to constant factors.