Robust Sequential Learning in Random Order Networks

TL;DR

The paper studies sequential Bayesian learning on networks under uniformly random decision orders and proves that random-order learning is robust to adversarial graph modifications, unlike fragile strategic-order learning. It establishes a monotonicity-based bound showing that a vertex with learning rate $1-\varepsilon$ maintains at least $1-(k+1)\varepsilon$ learning after $k$ modifications, implying near-optimal tolerance $o(1/\varepsilon)$ to perturbations. It then builds practical network constructions that realize random-order learning, including boosting a complete graph with guinea pigs and embedding learning structures, and introduces a polynomial-time BoostGraph-MonteCarlo algorithm with submodular optimization guarantees to modify arbitrary networks toward random-order learning. The work connects topology to learnability, provides algorithmic tools for robust network design, and discusses computational challenges related to learning-oracle computations and potential hardness results for random-order learning rate estimation.

Abstract

In the sequential learning problem, agents in a network attempt to predict a binary ground truth, informed by both a noisy private signal and the predictions of neighboring agents before them. It is well known that social learning in this setting can be highly fragile: small changes to the action ordering, network topology, or even the strength of the agents' private signals can prevent a network from converging to the truth. We study networks that achieve random-order asymptotic truth learning, in which almost all agents learn the ground truth when the decision ordering is selected uniformly at random. We analyze the robustness of these networks, showing that those achieving random-order asymptotic truth learning are resilient to a bounded number of adversarial modifications. We characterize necessary conditions for such networks to succeed in this setting and introduce several graph constructions that learn through different mechanisms. Finally, we present a randomized polynomial-time algorithm that transforms an arbitrary network into one achieving random-order learning using minimal edge or vertex modifications, with provable approximation guarantees. Our results reveal structural properties of networks that achieve random-order learning and provide algorithmic tools for designing robust social networks.

Figures (5)

• Figure 1: A network that obtains learning for $q < q_0$ but not for $q > q_0$, for some $q_0 \in [\frac{1}{2}, 1]$. The edges in the graph are oriented to indicate the direction of information flow.
• Figure 2: The celebrity graph as described in Bahar2020-am
• Figure 3: (a): By equipping a small number of vertices in a $K_n$ graph with many guinea pigs, the entire network can be boosted to achieve learning. (b): Example of $K_n$ being boosted by $\sqrt{n}$ complete binary trees of size $\log \log n$ to achieve random-order asymptotic learning.
• Figure 4: BoostAgents boosts the vertices $S \subseteq V$ by connecting them to an added set of celebrities $\{w_1, \cdots, w_k\}$.
• Figure 5: Network that obtains learning for $q > q_0$ but not for $q < q_0$. For each $i$, we have condensed the vertices $v_{i0}, \dots, v_{i4}$ into one gadget vertex $v_i$ for visual clarity.

Theorems & Definitions (27)

• Proposition 2.1: Generalized improvement principle
• Lemma 2.2
• Lemma 2.3
• Remark
• Proposition 3.1
• Lemma 3.2: Monotonicity
• Theorem 3.3
• Corollary 3.3.1
• Corollary 3.3.2
• Theorem 3.4