Optimal Networks for Agentic Information Aggregation

2026-09-28

AI Summary

This study investigates the exactness of information aggregation among agents in networked learning, characterizing the topological limits for achieving globally optimal predictions under both adaptive and oblivious settings. By modeling linear prediction error propagation over directed acyclic graphs and integrating combinatorial optimization with asymptotic analysis, we rigorously prove that single-parent structures cannot guarantee exact aggregation, whereas two parents suffice in the oblivious setting. Furthermore, we derive tight theoretical bounds on network depth and agent count, constructing optimal network architectures with depth O(d log d) or O(d) and agent complexity O(d²), which achieve theoretical optimality up to constant factors.

Abstract

We study information aggregation in the networked learning model introduced by Kearns, Roth, and Ryu (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, Roth, and Ryu 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 $d\ge3$, 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.

Problem

• information aggregation
• networked learning
• directed acyclic graph
• linear predictor
• optimal network design

Innovation

• Information Aggregation
• Networked Learning
• Agentic Networks
• Optimal Network Design
• Linear Prediction