Lower Bounds on Active Learning for Graphical Model Selection
We consider the problem of estimating the underlying graph associated with a Markov random field, with the added twist that the decoding algorithm can iteratively choose which subsets of nodes to sample based on the previous samples, i.e., active learning. Considering both Ising and Gaussian models, we provide algorithm-independent lower bounds for high-probability recovery within the class of degree-bounded graphs. Our main results are minimax lower bounds for the active setting that match the best known lower bounds for the passive setting, which are known to be tight in several cases. Our analysis is based on a novel variation of Fano's inequality for the active learning setting. While we consider graph ensembles that are similar or identical to those considered for the passive setting, we require different techniques to analyze them, with the key challenge being bounding a mutual information quantity associated with observed subsets of the nodes, as opposed to full observations.