Bayesian Recovery for Probabilistic Coalition Structures

TL;DR

This paper demonstrates that Sparse Bayesian Learning outperforms traditional sparse recovery methods like l1 relaxations and greedy pursuits in recovering optimal coalition structures in probabilistic coalition structure generation, especially under high coherence conditions.

Contribution

It proves that SBL with a Gaussian-Gamma hierarchy is support consistent in PCSG, unlike standard methods which fail under high coherence.

Findings

l1 relaxations and greedy pursuits fail in high-coherence PCSG regimes.

SBL with Gaussian-Gamma hierarchy reliably recovers true coalition support.

Supports a separation between Bayesian and convex/greedy sparse methods in PCSG.

Abstract

Probabilistic Coalition Structure Generation (PCSG) is NP-hard and can be recast as an $l_0$-type sparse recovery problem by representing coalition structures as sparse coefficient vectors over a coalition-incidence design. A natural question is whether standard sparse methods, such as $l_1$ relaxations and greedy pursuits, can reliably recover the optimal coalition structure in this setting. We show that the answer is negative in a PCSG-inspired regime where overlapping coalitions generate highly coherent, near-duplicate columns: the irrepresentable condition fails for the design, and $k$-step Orthogonal Matching Pursuit (OMP) exhibits a nonvanishing probability of irreversible mis-selection. In contrast, we prove that Sparse Bayesian Learning (SBL) with a Gaussian-Gamma hierarchy is support consistent under the same structural assumptions. The concave sparsity penalty induced by SBL suppresses spurious near-duplicates and recovers the true coalition support with probability tending to one. This establishes a rigorous separation between convex, greedy, and Bayesian sparse approaches for PCSG.