Optimal Lower Bounds for Networked Information Aggregation

The problem of networked information aggregation, studied in Kearns et al. (2026), involves a group of learners situated on the vertices of a directed acyclic graph $G$, each learning a linear predictor $\widehat Y$ for a fixed random variable $Y$ given access to a local feature, as well as the predictors learnt by its parents. Learning proceeds iteratively, with learners ordered according to a topological sort of $G$. The main quantity of interest is the error incurred by the current learner, constrained to this flow of information, with respect to the best linear predictor using all the features seen so far. When the studied error is the MSE, i.e., $\mathbb{E} (\widehat Y - Y)^2$, Kearns et al. (2026) show that the error is at most $O(1/\sqrt{D})$ along a path of length $D$. They also obtain a hard instance where the MSE is lower bounded by $Ω(1/D)$, leaving the correct order open. In this work, we resolve this central open problem, and obtain a family of worst case problem instances with a MSE lower bound of $Ω(1/\sqrt{D})$. By exploiting invariances in the structure of the learnt predictors, our analysis generalizes to all convex loss functions $\ell(\widehat Y, Y)$ satisfying regularity conditions which include strong convexity in a ball around the origin, and that the ideal predictor minimizing the population loss is positively correlated with the label. We show that networked information aggregation on a gaussian instance in our worst case family incurs an $\ell$-error lower bounded by $Ω(1/\sqrt{D})$ with respect to this ideal predictor. We demonstrate that a variety of common losses satisfy these regularity conditions. In particular, the logistic loss satisfies them, and hence our analysis also closes the gap between the upper and lower bounds in Bateni et al. (2026).

Computationally Efficient Collaborative Communication Via Regularity-Based Coarsening

Our results show that the existence of a short high-utility protocol already suffices for efficient communication. In particular, in a game with $n$ possible observations and $m$ actions: (1) For any achievable target utility $α$, we give an algorithm with $\mathrm{poly}(n, m, 1/ε)$ runtime that designs a protocol achieving utility at least $α-ε$ using only $2^{\mathcal O(CC_α(G))}/ε^2$ bits of communication. Here, $CC_α(G)$ is the minimum number of bits used by any protocol, even a computationally inefficient one, to achieve utility $α$. (2) We prove that this exponential dependence on $CC_α(G)$ is tight up to a constant. That is, unless $\mathrm P=\mathrm{NP}$, no polynomial-time algorithm can in general find optimal protocols using fewer than $2^{CC_α(G) -2}$ bits. We note that our results strictly weaken the assumptions required by prior work in the multi-agent information aggregation literature, filling a gap that had remained elusive even for games with constant $CC_α(G)$. In particular, prior guarantees for agreement-based information aggregation rely on structural assumptions such as informational substitutes or weak learnability. We show that these assumptions already imply $CC_α(G) = O(1)$ and are therefore more restrictive conditions than required by our protocol to succeed. On a technical level, our results involve a novel strengthening of the Frieze-Kannan weak regularity lemma and yield the following powerful polynomial-time transformation tool: for every communication game $G$, it constructs a game $\hat G$ that is a coarsening of the agents' observation spaces into constant-size partitions, such that $G$ and $\hat G$ are indistinguishable with respect to every short communication protocol. This coarsening theorem is the engine behind our algorithm and may be of independent interest.