Support Consistency of Direct Sparse-Change Learning in Markov Networks

Abstract: We study the problem of learning sparse structure changes between two Markov networks $P$ and $Q$. Rather than fitting two Markov networks separately to two sets of data and figuring out their differences, a recent work proposed to learn changes \emph{directly} via estimating the ratio between two Markov network models. In this paper, we give sufficient conditions for \emph{successful change detection} with respect to the sample size $n_p, n_q$, the dimension of data $m$, and the number of changed edges $d$. When using an unbounded density ratio model we prove that the true sparse changes can be consistently identified for $$n_p = \Omega(d² \log \frac{m<sup>2+m}{2})$$ and $n_q = \Omega({n_p<sup>2})$, with an exponentially decaying upper-bound on learning error. Such sample complexity can be improved to $$\min(n_p, n_q) = \Omega(d² \log \frac{m<sup>2+m}{2})$$ when the boundedness of the density ratio model is assumed. Our theoretical guarantee can be applied to a wide range of discrete/continuous Markov networks.