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In this paper we investigate the computational complexity of learning the graph structure underlying a discrete undirected graphical model from i.i.d.

In this paper we investigate the computational complexity of learning the graph structure underlying a discrete undirected graphical model from i.i.d.

In this paper we investigate the computational complexity of learning the graph structure underlying a discrete undirected graphical model from i.i.d. Our first result is an unconditional computational lower bound of $\Omega (p {d/2})$ for learning general graphical models on $p$ nodes of maximum degree $d$, for the class of statistical algorithms recently introduced by Feldman et al. The construction is related to the notoriously difficult learning parities with noise problem in computational learning theory. Our lower bound shows that the $\widetilde O(p {d 2})$ runtime required by Bresler, Mossel, and Sly's exhaustive-search algorithm cannot be significantly improved without restricting the class of models. Aside from structural assumptions on the graph such as it being a tree, hypertree, tree-like, etc., most recent papers on structure learning assume that the model has the correlation decay property.

In this paper we investigate the computational complexity of learning the graph structure underlying a discrete undirected graphical model from i.i.d. samples. Our first result is an unconditional computational lower bound of $\Omega (p^{d/2})$ for learning general graphical models on $p$ nodes of maximum degree $d$, for the class of statistical algorithms recently introduced by Feldman et al. The construction is related to the notoriously difficult learning parities with noise problem in computational learning theory. Our lower bound shows that the $\widetilde O(p^{d+2})$ runtime required by Bresler, Mossel, and Sly's exhaustive-search algorithm cannot be significantly improved without restricting the class of models. Aside from structural assumptions on the graph such as it being a tree, hypertree, tree-like, etc., most recent papers on structure learning assume that the model has the correlation decay property. Indeed, focusing on ferromagnetic Ising models, Bento and Montanari showed that all known low-complexity algorithms fail to learn simple graphs when the interaction strength exceeds a number related to the correlation decay threshold. Our second set of results gives a class of repelling (antiferromagnetic) models that have the \emph{opposite} behavior: very strong repelling allows efficient learning in time $\widetilde O(p^2)$. We provide an algorithm whose performance interpolates between $\widetilde O(p^2)$ and $\widetilde O(p^{d+2})$ depending on the strength of the repulsion.

In this paper we investigate the computational complexity of learning the graph structure underlying a discrete undirected graphical model from i.i.d. samples. We first observe that the notoriously difficult problem of learning parities with noise can be captured as a special case of learning graphical models. This leads to an unconditional computational lower bound of $\Omega (p^{d/2})$ for learning general graphical models on $p$ nodes of maximum degree $d$, for the class of so-called statistical algorithms recently introduced by Feldman et al (2013). The lower bound suggests that the $O(p^d)$ runtime required to exhaustively search over neighborhoods cannot be significantly improved without restricting the class of models. Aside from structural assumptions on the graph such as it being a tree, hypertree, tree-like, etc., many recent papers on structure learning assume that the model has the correlation decay property. Indeed, focusing on ferromagnetic Ising models, Bento and Montanari (2009) showed that all known low-complexity algorithms fail to learn simple graphs when the interaction strength exceeds a number related to the correlation decay threshold. Our second set of results gives a class of repelling (antiferromagnetic) models that have the opposite behavior: very strong interaction allows efficient learning in time $O(p^2)$. We provide an algorithm whose performance interpolates between $O(p^2)$ and $O(p^{d+2})$ depending on the strength of the repulsion.