Necessary Conditions for Consistent Set-Based Graphical Model Selection

Graphical model selection, where the goal is to estimate the graph underlying a distribution, is known to be an NP-hard problem. An important issue is to study theoretical limits on the performance of graphical model selection algorithms. In particular, given parameters of the underlying distribution, we want to find a lower bound on the number of samples required for accurate graph estimation. When deriving these theoretical bounds, it is common to treat the learning problem as a communication problem where the observations correspond to noisy messages and the decoding problem infers the graph from the observations. Current analysis of graphical model selection algorithms is limited to studying graph estimators that output a unique graph. In this paper, we consider graph estimators that output a set of graphs, leading to set-based graphical model selection (SB-GMS). This has connections to list-decoding where a decoder outputs a list of possible codewords instead of a single codeword. Our main contribution is to derive necessary conditions for accurate SB-GMS for various classes of graphical models and show reduction in the number of samples required for consistent estimation. Further, we derive necessary conditions on the cardinality of the set-based estimates given graph parameters.