Sparse Logistic Regression Learns All Discrete Pairwise Graphical Models
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We characterize the effectiveness of a natural and classic algorithm for recovering the Markov graph of a general discrete pairwise graphical model from i.i.d. samples. The algorithm is (appropriately regularized) conditional maximum likelihood, which involves solving a convex program for each node; for Ising models this is ℓ_1-constrained logistic regression, while for more alphabets an ℓ_2,1 group-norm constraint needs to be used. We show that this algorithm can recover any arbitrary discrete pairwise graphical model, and also characterize its sample complexity as a function of model width, alphabet size, edge parameter accuracy, and the number of variables. We show that along every one of these axes, it matches or improves on all existing results and algorithms for this problem. Our analysis applies a sharp generalization error bound for logistic regression when the weight vector has an ℓ_1 constraint (or ℓ_2,1 constraint) and the sample vector has an ℓ_∞ constraint (or ℓ_2, ∞ constraint). We also show that the proposed convex programs can be efficiently optimized in Õ(n^2) running time (where n is the number of variables) under the same statistical guarantee. Our experimental results verify our analysis.
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